Polymath Wiki - User contributions [en]

Bounded gaps between primes

2015-12-07T11:13:36Z

AndrewVSutherland: /* Recent papers and notes */

Finding narrow admissible tuples

2015-07-12T15:57:06Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel hybrid ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The large sieve inequality (in the sharp form of Selberg) [IK2004, Theorem 7.14] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54516.txt 54,516]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,586]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-Schinzel hybrid
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|Large sieve
| 28,080,008
| 2,342,970
| 1,184,955
| 197,097
| 145,712
| 128,972
| 126,932
| 55,179
| 35,236
| 30,983
| 26,389
| 24,038
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-Schinzel hybrid
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 2,771
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 2,765
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 2,633
| 1,361
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 2,631
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 2,338
| 1,214
|-
|Large sieve
| 18,860
| 15,784
| 13,697
| 8,616
| 7,548
| 3,960
| 2,559
| 2,393
| 2,273
| 1,192
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2015-07-12T15:56:25Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel hybrid ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The large sieve inequality (in the sharp form of Selberg) [IK2004, Theorem 7.14] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54516.txt 54,516]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,586]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-Schinzel hybrid
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|Large sieve
| 28,080,008
| 2,342,970
| 1,184,955
| 197,097
| 145,712
| 128,972
| 126,932
| 55,179
| 35,236
| 30,983
| 26,389
| 24,038
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 2,771
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 2,765
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 2,633
| 1,361
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 2,631
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 2,338
| 1,214
|-
|Large sieve
| 18,860
| 15,784
| 13,697
| 8,616
| 7,548
| 3,960
| 2,559
| 2,393
| 2,273
| 1,192
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2015-07-12T15:55:58Z

AndrewVSutherland: /* Greedy-Schinzel hybrid */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-Schinzel hybrid ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The large sieve inequality (in the sharp form of Selberg) [IK2004, Theorem 7.14] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54516.txt 54,516]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,586]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|Large sieve
| 28,080,008
| 2,342,970
| 1,184,955
| 197,097
| 145,712
| 128,972
| 126,932
| 55,179
| 35,236
| 30,983
| 26,389
| 24,038
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 2,771
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 2,765
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 2,633
| 1,361
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 2,631
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 2,338
| 1,214
|-
|Large sieve
| 18,860
| 15,784
| 13,697
| 8,616
| 7,548
| 3,960
| 2,559
| 2,393
| 2,273
| 1,192
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Bounded gaps between primes

2015-07-12T14:37:25Z

AndrewVSutherland: Added slides from a lecture series given at the Oberwolfach "Explicit Methods in Number Theory" workshop

Timeline of prime gap bounds

2015-07-05T22:38:25Z

AndrewVSutherland: Clarified the meaning of * in the legend

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| Aug 10 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| May 14 2013
| 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| May 21
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| May 28
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| May 30
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| May 31
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| Jun 1
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| Jun 2
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| Jun 3
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| Jun 4
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| Jun 5
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| Jun 6
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| Jun 7
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
5 [EH] ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493442.txt 493,442] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252987 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484192.txt 484,192] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252989 Sutherland])
|
|-
| Nov 25
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpinull_385.mpl 385]#*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_339.mpl 339]*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])
| [https://math.mit.edu/~drew/admissible_42392_484176.txt 484,176] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253019 Sutherland])

[http://math.mit.edu/~drew/admissible_43134_493436.txt 493,436][m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253086 Sutherland])
| Using the exponential moment method to control errors
|-
| Nov 26
|
| 102# ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253225 Nielsen])
| [http://math.mit.edu/~drew/admissible_43134_493426.txt 493,426] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253143 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484168_-209744.txt 484,168] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253160 xfxie])

[http://math.mit.edu/~primegaps/tuples/admissible_102_576.txt 576]# ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimising the original Maynard variational problem
|-
| Nov 27
|
|
| [https://math.mit.edu/~drew/admissible_42392_484162.txt 484,162] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253278 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484142.txt 484,142] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253381 Sutherland])
|
|-
| Nov 28
|
|
| [http://math.mit.edu/~drew/admissible_42392_484136.txt 484,136] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253621 Sutherland]

[http://math.mit.edu/~drew/admissible_42392_484126.txt 484,126] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253661 Sutherland])
|
|-
| Dec 4
|
| 64#? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255577 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_64_330.txt 330]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Searching over a wider range of polynomials than in Maynard's paper
|-
| Dec 6
|
|
| [http://math.mit.edu/~drew/admissible_43134_493408.txt 493,408] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-255735 Sutherland])
|
|-
| Dec 19
|
| 59#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257786 Nielsen])
10,000,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257821 Tao])

1,700,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])

38,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_59_300.txt 300]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
182,087,080? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257826 Sutherland])

179,933,380? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257833 Sutherland])
| More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the <math>M_k \geq \frac{k}{k-1} \log k - O(1)</math> lower bound.
|-
| Dec 20
|
| <strike>25,819? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257957 Castryck])</strike>
55#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])

36,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258079 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_35146_m2.mpl 35,146]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258090 xfxie])
| [http://math.mit.edu/~drew/admissible_10000000_175225874.txt 175,225,874]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])
27,398,976? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_26682014.txt 26,682,014]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257911 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_431682.txt 431,682]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257914 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_430448.txt 430,448]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257918 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_429822.txt 429,822]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_25819_283242.txt 283,242]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257960 Sutherland])</strike>

[http://math.mit.edu/~primegaps/tuples/admissible_55_272.txt 272]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
|
|-
| Dec 21
|
| [http://math.mit.edu/~drew/maple_3_1640042.txt 1,640,042]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258151 Sutherland])

<strike>[http://math.mit.edu/~drew/maple_4_41862295.txt 41,862,295]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland)]</strike>

[http://math.mit.edu/~drew/maple_3_1631027.txt 1,631,027]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1630680_m3.mpl 1,630,680]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258196 xfxie])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_36000000_m4.mpl 36,000,000]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258197 xfxie]</strike>

<strike>35,127,242? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_25589558_m4.mpl 25,589,558]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258250 xfxie])</strike>
| [http://math.mit.edu/~drew/admissible_38000_429798.txt 429,798]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_25602438.txt 25,602,438]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_36000_405528.txt 405,528]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258148 Sutherland])

<strike>825,018,354? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1631027_25533684.txt 25,533,684]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://math.mit.edu/~drew/admissible_35146_395264.txt 395,264]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_35146_395234_-190558.txt 395,234]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258194 xfxie])

[http://math.mit.edu/~drew/admissible_35146_395178.txt 395,178]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258198 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_25527718.txt 25,527,718]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258200 Sutherland])

<strike>685,833,596? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>491,149,914? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258267 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1630680_24490758.txt 24,490,758]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258268 Sutherland])
| Optimising the explicit lower bound <math>M_k \geq \log k-O(1)</math>
|-
| Dec 22
|
| 1,628,944? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

75,000,000? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

3,400,000,000? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

5,511 [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_74487363_m4.mpl 74,487,363]? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1628943_m3.mpl 1,628,943]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])
| [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_24490410.txt 24,490,410]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

<strike>485,825,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_35146_395122.txt 395,122]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])

<strike>473,244,502? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])</strike>

1,523,781,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

82,575,303,678? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52130.txt 52,130]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258452 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])
| A numerical precision issue was discovered in the earlier m=4 calculations
|-
| Dec 23
|
| 41,589? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_41588_m4EH.mpl 41,588]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258555 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_309661_m5EH.mpl 309,661]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258558 xfxie])

[http://math.mit.edu/~drew/maple_4_BV.txt 105,754,838]#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[https://math.mit.edu/~drew/maple_5_BV.txt 5,300,000,000]#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
| [http://math.mit.edu/~drew/admissible_1628943_24462774.txt 24,462,774]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

1,512,832,950? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4146936.txt 4,146,936]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52116.txt 52,116] [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_41589_474600.txt 474,600]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474460.txt 474,460]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258569 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4143140.txt 4,143,140]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258570 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313942.txt 32,313,942]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258572 Sutherland])

2,186,561,568#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474372.txt 474,372]? [EH] [m=4]
([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258609 Sutherland])

131,161,149,090#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
|
|-
| Dec 24
|
|
| [http://math.mit.edu/~drew/admissible_41588_474320.txt 474,320]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137872.txt 4,137,872]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_1628943_24462654.txt 24,462,654]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

1,497,901,734? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313878.txt 32,313,878]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])
|
|-
| Dec 28
|
|
| [http://math.mit.edu/~drew/admissible_41588_474296.txt 474,296]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137854.txt 4,137,854] [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])
|
|-
| Jan 2 2014
|
|
| [http://math.mit.edu/~drew/admissible_41588_474290.txt 474,290]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-260937 Sutherland])
|
|-
| Jan 6
|
| 54# ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen])
| 270# ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis])
|
|-
| Jan 8
|
| 4 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| 8 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262511 here].
|-
| Jan 9
|
|
| [http://math.mit.edu/~drew/admissible_41588_474266.txt 474,266] [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])
|
|-
| Jan 28
|
|
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland])
|
|-
| Jan 29
|
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
|-
| Feb 9
|
|
|
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here]
|-
| Feb 17
|
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen])
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis])
| Managed to get the epsilon trick to be computationally feasible for medium k
|-
| Feb 22
|
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen])
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| More efficient matrix computation allows for higher degrees to be used
|-
| Mar 4
|
|
|
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed]
|-
| Apr 14
|
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen])
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis])
| A 2-week computer calculation!
|-
| Apr 17
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410] [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821] [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707] [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908] [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
|398,646? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

25,816,462? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

1,541,858,666? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

84,449,123,072? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones
|-
| Apr 18
|
|
| [http://math.mit.edu/~drew/admissible_35410_398244.txt 398,244]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24798306.txt 24,798,306]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

1,541,183,756? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

84,449,103,908? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])
|
|-
| Apr 28
|
|
| [http://math.mit.edu/~drew/admissible_35410_398130.txt 398,130] [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24797814.txt 24,797,814] [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

1,526,698,470? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

83,833,839,882? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])
|-
| May 1
|
|
| 81,973,172,502? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])
2,165,674,446#? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])

130,235,143,908#? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])
| faster admissibility testing
|-
| May 3
|
|
| 1,460,493,420? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])
80,088,836,006? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

1,488,227,220?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

81,912,638,914?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

2,111,605,786?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

127,277,395,046?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])
| Fast admissibility testing for Hensley-Richards tuples
|-
| May 3
|
| 3,393,468,735? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])
2,113,163?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])

105,754,479?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])

5,274,206,963?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])
|
| Improved hillclimbing; also confirmation of previous k values
|-
| May 4
|
|
| 79,929,339,154? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])
[http://math.mit.edu/~drew/admissible_2113163_32588668.txt 32,588,668]?#* [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])

2,111,597,632?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])

126,630,432,986?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])
|-
| May 5
|
|
| [http://math.mit.edu/~drew/admissible_2113163_32285928.txt 32,285,928]?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-324263 Sutherland])
|-
| May 9
|
|
| 1,460,485,532? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])
79,929,332,990? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

1,488,222,198?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

81,912,604,302?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

2,111,417,340?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

126,630,386,774?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])
| Fast admissibility testing for Hensley-Richards sequences
|-
| May 14
|
|
| 1,440,495,268? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])
78,807,316,822 [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

1,467,584,468?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

80,761,835,464?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

2,082,729,956?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

124,840,189,042?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])
| Fast admissibility testing for Schinzel sequences
|-
| May 18
|
|
| 1,435,011,318? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])
1,462,568,450?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])

2,075,186,584?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])
| Faster modified Schinzel sieve testing
|-
| May 23
|
|
| 1,424,944,070? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland])
1,452,348,402?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland])
| Fast restricted greedy sieving
|-
| May 28
|
| 52? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355568 de Grey])
51? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355656 de Grey])
| 254? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_52_254.txt Clark-Jarvis])
252? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| New bounds for <math>M_{k,1/(k-1)}</math>
|-
| May 30
|
|
| [http://math.mit.edu/~drew/greedy_74487363_1404556152.txt 1,404,556,152]? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])
[http://math.mit.edu/~drew/greedy_75845707_1431556072.txt 1,431,556,072]* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])

[http://math.mit.edu/~drew/greedy_105754837_2031558336.txt 2,031,558,336]?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])
| Heuristically determined shift for the shifted greedy sieve
|-
| June 8
|
|
| [http://math.mit.edu/~drew/schinzel_3473955908_80550202480.txt 80,550,202,480]* [m=5] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-366807 Sutherland])
| Verification of several previous bounds
|-
| June 23
|
|
| [http://math.mit.edu/~drew/schinzel_3393468735_78602310160.txt 78,602,310,160]? [m=5] ([http://terrytao.wordpress.com/2014/06/19/polymath8-wrapping-up/#comment-378098 Sutherland])
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent or conditional result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
# [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

Bounded gaps between primes

2015-07-01T19:58:43Z

AndrewVSutherland: /* Recent papers and notes */

Timeline of prime gap bounds

2014-06-08T17:56:22Z

AndrewVSutherland:

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| Aug 10 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| May 14 2013
| 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| May 21
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| May 28
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| May 30
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| May 31
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| Jun 1
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| Jun 2
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| Jun 3
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| Jun 4
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| Jun 5
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| Jun 6
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| Jun 7
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
5 [EH] ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493442.txt 493,442] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252987 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484192.txt 484,192] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252989 Sutherland])
|
|-
| Nov 25
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpinull_385.mpl 385]#*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_339.mpl 339]*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])
| [https://math.mit.edu/~drew/admissible_42392_484176.txt 484,176] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253019 Sutherland])

[http://math.mit.edu/~drew/admissible_43134_493436.txt 493,436][m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253086 Sutherland])
| Using the exponential moment method to control errors
|-
| Nov 26
|
| 102# ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253225 Nielsen])
| [http://math.mit.edu/~drew/admissible_43134_493426.txt 493,426] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253143 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484168_-209744.txt 484,168] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253160 xfxie])

[http://math.mit.edu/~primegaps/tuples/admissible_102_576.txt 576]# ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimising the original Maynard variational problem
|-
| Nov 27
|
|
| [https://math.mit.edu/~drew/admissible_42392_484162.txt 484,162] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253278 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484142.txt 484,142] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253381 Sutherland])
|
|-
| Nov 28
|
|
| [http://math.mit.edu/~drew/admissible_42392_484136.txt 484,136] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253621 Sutherland]

[http://math.mit.edu/~drew/admissible_42392_484126.txt 484,126] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253661 Sutherland])
|
|-
| Dec 4
|
| 64#? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255577 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_64_330.txt 330]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Searching over a wider range of polynomials than in Maynard's paper
|-
| Dec 6
|
|
| [http://math.mit.edu/~drew/admissible_43134_493408.txt 493,408] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-255735 Sutherland])
|
|-
| Dec 19
|
| 59#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257786 Nielsen])
10,000,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257821 Tao])

1,700,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])

38,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_59_300.txt 300]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
182,087,080? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257826 Sutherland])

179,933,380? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257833 Sutherland])
| More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the <math>M_k \geq \frac{k}{k-1} \log k - O(1)</math> lower bound.
|-
| Dec 20
|
| <strike>25,819? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257957 Castryck])</strike>
55#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])

36,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258079 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_35146_m2.mpl 35,146]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258090 xfxie])
| [http://math.mit.edu/~drew/admissible_10000000_175225874.txt 175,225,874]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])
27,398,976? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_26682014.txt 26,682,014]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257911 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_431682.txt 431,682]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257914 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_430448.txt 430,448]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257918 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_429822.txt 429,822]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_25819_283242.txt 283,242]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257960 Sutherland])</strike>

[http://math.mit.edu/~primegaps/tuples/admissible_55_272.txt 272]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
|
|-
| Dec 21
|
| [http://math.mit.edu/~drew/maple_3_1640042.txt 1,640,042]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258151 Sutherland])

<strike>[http://math.mit.edu/~drew/maple_4_41862295.txt 41,862,295]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland)]</strike>

[http://math.mit.edu/~drew/maple_3_1631027.txt 1,631,027]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1630680_m3.mpl 1,630,680]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258196 xfxie])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_36000000_m4.mpl 36,000,000]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258197 xfxie]</strike>

<strike>35,127,242? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_25589558_m4.mpl 25,589,558]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258250 xfxie])</strike>
| [http://math.mit.edu/~drew/admissible_38000_429798.txt 429,798]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_25602438.txt 25,602,438]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_36000_405528.txt 405,528]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258148 Sutherland])

<strike>825,018,354? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1631027_25533684.txt 25,533,684]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://math.mit.edu/~drew/admissible_35146_395264.txt 395,264]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_35146_395234_-190558.txt 395,234]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258194 xfxie])

[http://math.mit.edu/~drew/admissible_35146_395178.txt 395,178]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258198 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_25527718.txt 25,527,718]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258200 Sutherland])

<strike>685,833,596? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>491,149,914? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258267 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1630680_24490758.txt 24,490,758]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258268 Sutherland])
| Optimising the explicit lower bound <math>M_k \geq \log k-O(1)</math>
|-
| Dec 22
|
| 1,628,944? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

75,000,000? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

3,400,000,000? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

5,511? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_74487363_m4.mpl 74,487,363]? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1628943_m3.mpl 1,628,943]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])
| [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_24490410.txt 24,490,410]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

<strike>485,825,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_35146_395122.txt 395,122]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])

<strike>473,244,502? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])</strike>

1,523,781,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

82,575,303,678? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52130.txt 52,130]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258452 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])
| A numerical precision issue was discovered in the earlier m=4 calculations
|-
| Dec 23
|
| 41,589? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_41588_m4EH.mpl 41,588]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258555 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_309661_m5EH.mpl 309,661]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258558 xfxie])

[http://math.mit.edu/~drew/maple_4_BV.txt 105,754,838]#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[https://math.mit.edu/~drew/maple_5_BV.txt 5,300,000,000]#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
| [http://math.mit.edu/~drew/admissible_1628943_24462774.txt 24,462,774]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

1,512,832,950? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4146936.txt 4,146,936]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52116.txt 52,116]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_41589_474600.txt 474,600]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474460.txt 474,460]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258569 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4143140.txt 4,143,140]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258570 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313942.txt 32,313,942]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258572 Sutherland])

2,186,561,568#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474372.txt 474,372]? [EH] [m=4]
([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258609 Sutherland])

131,161,149,090#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
|
|-
| Dec 24
|
|
| [http://math.mit.edu/~drew/admissible_41588_474320.txt 474,320]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137872.txt 4,137,872]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_1628943_24462654.txt 24,462,654]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

1,497,901,734? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313878.txt 32,313,878]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])
|
|-
| Dec 28
|
|
| [http://math.mit.edu/~drew/admissible_41588_474296.txt 474,296]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137854.txt 4,137,854]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])
|
|-
| Jan 2 2014
|
|
| [http://math.mit.edu/~drew/admissible_41588_474290.txt 474,290]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-260937 Sutherland])
|
|-
| Jan 6
|
| 54# ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen])
| 270# ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis])
|
|-
| Jan 8
|
| 4 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| 8 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262511 here].
|-
| Jan 9
|
|
| [http://math.mit.edu/~drew/admissible_41588_474266.txt 474,266]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])
|
|-
| Jan 28
|
|
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland])
|
|-
| Jan 29
|
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
|-
| Feb 9
|
|
|
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here]
|-
| Feb 17
|
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen])
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis])
| Managed to get the epsilon trick to be computationally feasible for medium k
|-
| Feb 22
|
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen])
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| More efficient matrix computation allows for higher degrees to be used
|-
| Mar 4
|
|
|
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed]
|-
| Apr 14
|
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen])
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis])
| A 2-week computer calculation!
|-
| Apr 17
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707]? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908]? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
|398,646? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

25,816,462? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

1,541,858,666? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

84,449,123,072? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones
|-
| Apr 18
|
|
| [http://math.mit.edu/~drew/admissible_35410_398244.txt 398,244]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24798306.txt 24,798,306]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

1,541,183,756? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

84,449,103,908? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])
|
|-
| Apr 28
|
|
| [http://math.mit.edu/~drew/admissible_35410_398130.txt 398,130]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24797814.txt 24,797,814]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

1,526,698,470? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])

83,833,839,882? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland])
|-
| May 1
|
|
| 81,973,172,502? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])
2,165,674,446#? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])

130,235,143,908#? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland])
| faster admissibility testing
|-
| May 3
|
|
| 1,460,493,420? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])
80,088,836,006? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

1,488,227,220?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

81,912,638,914?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

2,111,605,786?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])

127,277,395,046?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])
| Fast admissibility testing for Hensley-Richards tuples
|-
| May 3
|
| 3,393,468,735? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])
2,113,163?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])

105,754,479?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])

5,274,206,963?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey])
|
| Improved hillclimbing; also confirmation of previous k values
|-
| May 4
|
|
| 79,929,339,154? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])
[http://math.mit.edu/~drew/admissible_2113163_32588668.txt 32,588,668]?#* [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])

2,111,597,632?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])

126,630,432,986?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland])
|-
| May 5
|
|
| [http://math.mit.edu/~drew/admissible_2113163_32285928.txt 32,285,928]?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-324263 Sutherland])
|-
| May 9
|
|
| 1,460,485,532? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])
79,929,332,990? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

1,488,222,198?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

81,912,604,302?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

2,111,417,340?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])

126,630,386,774?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland])
| Fast admissibility testing for Hensley-Richards sequences
|-
| May 14
|
|
| 1,440,495,268? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])
78,807,316,822 [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

1,467,584,468?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

80,761,835,464?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

2,082,729,956?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])

124,840,189,042?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland])
| Fast admissibility testing for Schinzel sequences
|-
| May 18
|
|
| 1,435,011,318? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])
1,462,568,450?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])

2,075,186,584?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland])
| Faster modified Schinzel sieve testing
|-
| May 23
|
|
| 1,424,944,070? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland])
1,452,348,402?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland])
| Fast restricted greedy sieving
|-
| May 28
|
| 52? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355568 de Grey])
51? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355656 de Grey])
| 254? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_52_254.txt Clark-Jarvis])
252? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| New bounds for <math>M_{k,1/(k-1)}</math>
|-
| May 30
|
|
| [http://math.mit.edu/~drew/greedy_74487363_1404556152.txt 1,404,556,152]? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])
[http://math.mit.edu/~drew/greedy_75845707_1431556072.txt 1,431,556,072]?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])

[http://math.mit.edu/~drew/greedy_105754837_2031558336.txt 2,031,558,336]?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland])
| Heuristically determined shift for the shifted greedy sieve
|-
| June 8
|
|
| [http://math.mit.edu/~drew/schinzel_3473955908_80550202480.txt 80,550,202,480]* [m=5] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-366807 Sutherland])
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
# [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

Bounded gaps between primes

2014-05-23T18:03:53Z

AndrewVSutherland: /* Code and data */

Bounded gaps between primes

2014-05-21T22:42:55Z

AndrewVSutherland: /* Current records */

Bounded gaps between primes

2014-05-18T20:58:58Z

AndrewVSutherland: /* Code and data */

Bounded gaps between primes

2014-05-14T13:27:05Z

AndrewVSutherland: /* Code and data */

Bounded gaps between primes

2014-05-13T08:30:39Z

AndrewVSutherland: /* Recent papers and notes */

Bounded gaps between primes

2014-05-09T15:16:31Z

AndrewVSutherland: /* Code and data */

Bounded gaps between primes

2014-05-02T18:33:20Z

AndrewVSutherland: /* Code and data */

Bounded gaps between primes

2014-04-30T00:49:30Z

AndrewVSutherland: /* Code and data */

Finding narrow admissible tuples

2014-04-29T15:04:50Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The large sieve inequality (in the sharp form of Selberg) [IK2004, Theorem 7.14] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54516.txt 54,516]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,586]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|Large sieve
| 28,080,008
| 2,342,970
| 1,184,955
| 197,097
| 145,712
| 128,972
| 126,932
| 55,179
| 35,236
| 30,983
| 26,389
| 24,038
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 2,771
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 2,765
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 2,633
| 1,361
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 2,631
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 2,338
| 1,214
|-
|Large sieve
| 18,860
| 15,784
| 13,697
| 8,616
| 7,548
| 3,960
| 2,559
| 2,393
| 2,273
| 1,192
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2014-04-29T15:02:21Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The large sieve inequality (in the sharp form of Selberg) [IK2004, Theorem 7.14] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54516.txt 54,516]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|Large sieve
| 28,080,008
| 2,342,970
| 1,184,955
| 197,097
| 145,712
| 128,972
| 126,932
| 55,179
| 35,236
| 30,983
| 26,389
| 24,038
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 2,771
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 2,765
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 2,633
| 1,361
|-
|Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 2,631
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 2,338
| 1,214
|-
|Large sieve
| 18,860
| 15,784
| 13,697
| 8,616
| 7,548
| 3,960
| 2,559
| 2,393
| 2,273
| 1,192
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Bounded gaps between primes

2014-04-22T09:20:05Z

AndrewVSutherland: /* Recent papers and notes */

Timeline of prime gap bounds

2014-04-20T09:03:19Z

AndrewVSutherland:

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| Aug 10 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| May 14 2013
| 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| May 21
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| May 28
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| May 30
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| May 31
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| Jun 1
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| Jun 2
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| Jun 3
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| Jun 4
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| Jun 5
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| Jun 6
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| Jun 7
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
5 [EH] ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493442.txt 493,442] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252987 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484192.txt 484,192] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252989 Sutherland])
|
|-
| Nov 25
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpinull_385.mpl 385]#*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_339.mpl 339]*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])
| [https://math.mit.edu/~drew/admissible_42392_484176.txt 484,176] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253019 Sutherland])

[http://math.mit.edu/~drew/admissible_43134_493436.txt 493,436][m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253086 Sutherland])
| Using the exponential moment method to control errors
|-
| Nov 26
|
| 102# ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253225 Nielsen])
| [http://math.mit.edu/~drew/admissible_43134_493426.txt 493,426] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253143 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484168_-209744.txt 484,168] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253160 xfxie])

[http://math.mit.edu/~primegaps/tuples/admissible_102_576.txt 576]# ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimising the original Maynard variational problem
|-
| Nov 27
|
|
| [https://math.mit.edu/~drew/admissible_42392_484162.txt 484,162] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253278 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484142.txt 484,142] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253381 Sutherland])
|
|-
| Nov 28
|
|
| [http://math.mit.edu/~drew/admissible_42392_484136.txt 484,136] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253621 Sutherland]

[http://math.mit.edu/~drew/admissible_42392_484126.txt 484,126] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253661 Sutherland])
|
|-
| Dec 4
|
| 64#? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255577 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_64_330.txt 330]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Searching over a wider range of polynomials than in Maynard's paper
|-
| Dec 6
|
|
| [http://math.mit.edu/~drew/admissible_43134_493408.txt 493,408] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-255735 Sutherland])
|
|-
| Dec 19
|
| 59#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257786 Nielsen])
10,000,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257821 Tao])

1,700,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])

38,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_59_300.txt 300]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
182,087,080? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257826 Sutherland])

179,933,380? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257833 Sutherland])
| More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the <math>M_k \geq \frac{k}{k-1} \log k - O(1)</math> lower bound.
|-
| Dec 20
|
| <strike>25,819? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257957 Castryck])</strike>
55#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])

36,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258079 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_35146_m2.mpl 35,146]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258090 xfxie])
| [http://math.mit.edu/~drew/admissible_10000000_175225874.txt 175,225,874]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])
27,398,976? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_26682014.txt 26,682,014]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257911 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_431682.txt 431,682]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257914 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_430448.txt 430,448]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257918 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_429822.txt 429,822]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_25819_283242.txt 283,242]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257960 Sutherland])</strike>

[http://math.mit.edu/~primegaps/tuples/admissible_55_272.txt 272]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
|
|-
| Dec 21
|
| [http://math.mit.edu/~drew/maple_3_1640042.txt 1,640,042]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258151 Sutherland])

<strike>[http://math.mit.edu/~drew/maple_4_41862295.txt 41,862,295]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland)]</strike>

[http://math.mit.edu/~drew/maple_3_1631027.txt 1,631,027]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1630680_m3.mpl 1,630,680]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258196 xfxie])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_36000000_m4.mpl 36,000,000]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258197 xfxie]</strike>

<strike>35,127,242? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_25589558_m4.mpl 25,589,558]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258250 xfxie])</strike>
| [http://math.mit.edu/~drew/admissible_38000_429798.txt 429,798]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_25602438.txt 25,602,438]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_36000_405528.txt 405,528]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258148 Sutherland])

<strike>825,018,354? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1631027_25533684.txt 25,533,684]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://math.mit.edu/~drew/admissible_35146_395264.txt 395,264]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_35146_395234_-190558.txt 395,234]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258194 xfxie])

[http://math.mit.edu/~drew/admissible_35146_395178.txt 395,178]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258198 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_25527718.txt 25,527,718]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258200 Sutherland])

<strike>685,833,596? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>491,149,914? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258267 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1630680_24490758.txt 24,490,758]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258268 Sutherland])
| Optimising the explicit lower bound <math>M_k \geq \log k-O(1)</math>
|-
| Dec 22
|
| 1,628,944? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

75,000,000? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

3,400,000,000? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

5,511? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_74487363_m4.mpl 74,487,363]? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1628943_m3.mpl 1,628,943]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])
| [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_24490410.txt 24,490,410]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

<strike>485,825,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_35146_395122.txt 395,122]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])

<strike>473,244,502? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])</strike>

1,523,781,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

82,575,303,678? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52130.txt 52,130]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258452 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])
| A numerical precision issue was discovered in the earlier m=4 calculations
|-
| Dec 23
|
| 41,589? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_41588_m4EH.mpl 41,588]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258555 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_309661_m5EH.mpl 309,661]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258558 xfxie])

[http://math.mit.edu/~drew/maple_4_BV.txt 105,754,838]#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[https://math.mit.edu/~drew/maple_5_BV.txt 5,300,000,000]#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
| [http://math.mit.edu/~drew/admissible_1628943_24462774.txt 24,462,774]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

1,512,832,950? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4146936.txt 4,146,936]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52116.txt 52,116]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_41589_474600.txt 474,600]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474460.txt 474,460]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258569 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4143140.txt 4,143,140]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258570 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313942.txt 32,313,942]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258572 Sutherland])

2,186,561,568#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474372.txt 474,372]? [EH] [m=4]
([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258609 Sutherland])

131,161,149,090#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
|
|-
| Dec 24
|
|
| [http://math.mit.edu/~drew/admissible_41588_474320.txt 474,320]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137872.txt 4,137,872]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_1628943_24462654.txt 24,462,654]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

1,497,901,734? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313878.txt 32,313,878]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])
|
|-
| Dec 28
|
|
| [http://math.mit.edu/~drew/admissible_41588_474296.txt 474,296]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137854.txt 4,137,854]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])
|
|-
| Jan 2 2014
|
|
| [http://math.mit.edu/~drew/admissible_41588_474290.txt 474,290]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-260937 Sutherland])
|
|-
| Jan 6
|
| 54# ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen])
| 270# ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis])
|
|-
| Jan 8
|
| 4 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| 8 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262511 here].
|-
| Jan 9
|
|
| [http://math.mit.edu/~drew/admissible_41588_474266.txt 474,266]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])
|
|-
| Jan 28
|
|
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland])
|
|-
| Jan 29
|
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
|-
| Feb 9
|
|
|
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here]
|-
| Feb 17
|
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen])
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis])
| Managed to get the epsilon trick to be computationally feasible for medium k
|-
| Feb 22
|
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen])
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| More efficient matrix computation allows for higher degrees to be used
|-
| Mar 4
|
|
|
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed]
|-
| Apr 14
|
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen])
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis])
| A 2-week computer calculation!
|-
| Apr 17
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707]? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908]? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
|398,646? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

25,816,462? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

1,541,858,666? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

84,449,123,072? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones
|-
| Apr 18
|
|
| [http://math.mit.edu/~drew/admissible_35410_398244.txt 398,244]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24798306.txt 24,798,306]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

1,541,183,756? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

84,449,103,908? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])
|
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
# [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

Timeline of prime gap bounds

2014-04-20T08:59:35Z

AndrewVSutherland:

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| Aug 10 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| May 14 2013
| 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| May 21
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| May 28
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| May 30
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| May 31
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| Jun 1
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| Jun 2
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| Jun 3
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| Jun 4
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| Jun 5
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| Jun 6
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| Jun 7
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
5 [EH] ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493442.txt 493,442] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252987 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484192.txt 484,192] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252989 Sutherland])
|
|-
| Nov 25
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpinull_385.mpl 385]#*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_339.mpl 339]*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])
| [https://math.mit.edu/~drew/admissible_42392_484176.txt 484,176] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253019 Sutherland])

[http://math.mit.edu/~drew/admissible_43134_493436.txt 493,436][m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253086 Sutherland])
| Using the exponential moment method to control errors
|-
| Nov 26
|
| 102# ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253225 Nielsen])
| [http://math.mit.edu/~drew/admissible_43134_493426.txt 493,426] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253143 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484168_-209744.txt 484,168] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253160 xfxie])

[http://math.mit.edu/~primegaps/tuples/admissible_102_576.txt 576]# ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimising the original Maynard variational problem
|-
| Nov 27
|
|
| [https://math.mit.edu/~drew/admissible_42392_484162.txt 484,162] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253278 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484142.txt 484,142] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253381 Sutherland])
|
|-
| Nov 28
|
|
| [http://math.mit.edu/~drew/admissible_42392_484136.txt 484,136] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253621 Sutherland]

[http://math.mit.edu/~drew/admissible_42392_484126.txt 484,126] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253661 Sutherland])
|
|-
| Dec 4
|
| 64#? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255577 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_64_330.txt 330]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Searching over a wider range of polynomials than in Maynard's paper
|-
| Dec 6
|
|
| [http://math.mit.edu/~drew/admissible_43134_493408.txt 493,408] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-255735 Sutherland])
|
|-
| Dec 19
|
| 59#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257786 Nielsen])
10,000,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257821 Tao])

1,700,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])

38,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_59_300.txt 300]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
182,087,080? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257826 Sutherland])

179,933,380? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257833 Sutherland])
| More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the <math>M_k \geq \frac{k}{k-1} \log k - O(1)</math> lower bound.
|-
| Dec 20
|
| <strike>25,819? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257957 Castryck])</strike>
55#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])

36,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258079 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_35146_m2.mpl 35,146]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258090 xfxie])
| [http://math.mit.edu/~drew/admissible_10000000_175225874.txt 175,225,874]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])
27,398,976? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_26682014.txt 26,682,014]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257911 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_431682.txt 431,682]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257914 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_430448.txt 430,448]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257918 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_429822.txt 429,822]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_25819_283242.txt 283,242]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257960 Sutherland])</strike>

[http://math.mit.edu/~primegaps/tuples/admissible_55_272.txt 272]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
|
|-
| Dec 21
|
| [http://math.mit.edu/~drew/maple_3_1640042.txt 1,640,042]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258151 Sutherland])

<strike>[http://math.mit.edu/~drew/maple_4_41862295.txt 41,862,295]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland)]</strike>

[http://math.mit.edu/~drew/maple_3_1631027.txt 1,631,027]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1630680_m3.mpl 1,630,680]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258196 xfxie])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_36000000_m4.mpl 36,000,000]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258197 xfxie]</strike>

<strike>35,127,242? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_25589558_m4.mpl 25,589,558]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258250 xfxie])</strike>
| [http://math.mit.edu/~drew/admissible_38000_429798.txt 429,798]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_25602438.txt 25,602,438]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_36000_405528.txt 405,528]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258148 Sutherland])

<strike>825,018,354? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1631027_25533684.txt 25,533,684]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://math.mit.edu/~drew/admissible_35146_395264.txt 395,264]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_35146_395234_-190558.txt 395,234]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258194 xfxie])

[http://math.mit.edu/~drew/admissible_35146_395178.txt 395,178]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258198 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_25527718.txt 25,527,718]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258200 Sutherland])

<strike>685,833,596? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>491,149,914? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258267 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1630680_24490758.txt 24,490,758]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258268 Sutherland])
| Optimising the explicit lower bound <math>M_k \geq \log k-O(1)</math>
|-
| Dec 22
|
| 1,628,944? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

75,000,000? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

3,400,000,000? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

5,511? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_74487363_m4.mpl 74,487,363]? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1628943_m3.mpl 1,628,943]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])
| [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_24490410.txt 24,490,410]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

<strike>485,825,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_35146_395122.txt 395,122]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])

<strike>473,244,502? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])</strike>

1,523,781,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

82,575,303,678? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52130.txt 52,130]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258452 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])
| A numerical precision issue was discovered in the earlier m=4 calculations
|-
| Dec 23
|
| 41,589? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_41588_m4EH.mpl 41,588]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258555 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_309661_m5EH.mpl 309,661]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258558 xfxie])

[http://math.mit.edu/~drew/maple_4_BV.txt 105,754,838]#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[https://math.mit.edu/~drew/maple_5_BV.txt 5,300,000,000]#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
| [http://math.mit.edu/~drew/admissible_1628943_24462774.txt 24,462,774]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

1,512,832,950? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4146936.txt 4,146,936]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52116.txt 52,116]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_41589_474600.txt 474,600]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474460.txt 474,460]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258569 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4143140.txt 4,143,140]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258570 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313942.txt 32,313,942]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258572 Sutherland])

2,186,561,568#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474372.txt 474,372]? [EH] [m=4]
([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258609 Sutherland])

131,161,149,090#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
|
|-
| Dec 24
|
|
| [http://math.mit.edu/~drew/admissible_41588_474320.txt 474,320]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137872.txt 4,137,872]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_1628943_24462654.txt 24,462,654]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

1,497,901,734? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313878.txt 32,313,878]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])
|
|-
| Dec 28
|
|
| [http://math.mit.edu/~drew/admissible_41588_474296.txt 474,296]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137854.txt 4,137,854]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])
|
|-
| Jan 2 2014
|
|
| [http://math.mit.edu/~drew/admissible_41588_474290.txt 474,290]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-260937 Sutherland])
|
|-
| Jan 6
|
| 54# ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen])
| 270# ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis])
|
|-
| Jan 8
|
| 4 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| 8 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262511 here].
|-
| Jan 9
|
|
| [http://math.mit.edu/~drew/admissible_41588_474266.txt 474,266]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])
|
|-
| Jan 28
|
|
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland])
|
|-
| Jan 29
|
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
|-
| Feb 9
|
|
|
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here]
|-
| Feb 17
|
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen])
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis])
| Managed to get the epsilon trick to be computationally feasible for medium k
|-
| Feb 22
|
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen])
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| More efficient matrix computation allows for higher degrees to be used
|-
| Mar 4
|
|
|
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed]
|-
| Apr 14
|
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen])
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis])
| A 2-week computer calculation!
|-
| Apr 17
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707]? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908]? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
|398,646? [m=2]* v
25,816,462? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

1,541,858,666? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])

84,449,123,072? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland])
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones
|-
| Apr 18
|
|
| [http://math.mit.edu/~drew/admissible_35410_398244.txt 398,244]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

[http://math.mit.edu/~drew/admissible_1649821_24798306.txt 24,798,306]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

1,541,183,756? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])

84,449,103,908? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland])
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
# [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

Timeline of prime gap bounds

2014-04-18T01:35:28Z

AndrewVSutherland: Corrected typo in k0 value for m=5 without EH or Deligne, 5,300,000 should be 5,300,000,000

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| Aug 10 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| May 14 2013
| 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/articles/7954 Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| May 21
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| May 28
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| May 30
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| May 31
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| Jun 1
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| Jun 2
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| Jun 3
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| Jun 4
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| Jun 5
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| Jun 6
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| Jun 7
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
5 [EH] ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493442.txt 493,442] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252987 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484192.txt 484,192] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252989 Sutherland])
|
|-
| Nov 25
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpinull_385.mpl 385]#*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_339.mpl 339]*? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253005 xfxie])
| [https://math.mit.edu/~drew/admissible_42392_484176.txt 484,176] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253019 Sutherland])

[http://math.mit.edu/~drew/admissible_43134_493436.txt 493,436][m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253086 Sutherland])
| Using the exponential moment method to control errors
|-
| Nov 26
|
| 102# ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-253225 Nielsen])
| [http://math.mit.edu/~drew/admissible_43134_493426.txt 493,426] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253143 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484168_-209744.txt 484,168] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253160 xfxie])

[http://math.mit.edu/~primegaps/tuples/admissible_102_576.txt 576]# ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Optimising the original Maynard variational problem
|-
| Nov 27
|
|
| [https://math.mit.edu/~drew/admissible_42392_484162.txt 484,162] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253278 Sutherland])

[http://math.mit.edu/~drew/admissible_42392_484142.txt 484,142] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253381 Sutherland])
|
|-
| Nov 28
|
|
| [http://math.mit.edu/~drew/admissible_42392_484136.txt 484,136] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253621 Sutherland]

[http://math.mit.edu/~drew/admissible_42392_484126.txt 484,126] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-253661 Sutherland])
|
|-
| Dec 4
|
| 64#? ([http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255577 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_64_330.txt 330]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
| Searching over a wider range of polynomials than in Maynard's paper
|-
| Dec 6
|
|
| [http://math.mit.edu/~drew/admissible_43134_493408.txt 493,408] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-255735 Sutherland])
|
|-
| Dec 19
|
| 59#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257786 Nielsen])
10,000,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257821 Tao])

1,700,000? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])

38,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257867 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_59_300.txt 300]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
182,087,080? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257826 Sutherland])

179,933,380? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257833 Sutherland])
| More efficient memory management allows for an increase in the degree of the polynomials used; the m=2,3 results use an explicit version of the <math>M_k \geq \frac{k}{k-1} \log k - O(1)</math> lower bound.
|-
| Dec 20
|
| <strike>25,819? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257957 Castryck])</strike>
55#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])

36,000? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258079 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_35146_m2.mpl 35,146]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258090 xfxie])
| [http://math.mit.edu/~drew/admissible_10000000_175225874.txt 175,225,874]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])
27,398,976? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257910 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_26682014.txt 26,682,014]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257911 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_431682.txt 431,682]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257914 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_430448.txt 430,448]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257918 Sutherland])

[http://math.mit.edu/~drew/admissible_38000_429822.txt 429,822]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_25819_283242.txt 283,242]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257960 Sutherland])</strike>

[http://math.mit.edu/~primegaps/tuples/admissible_55_272.txt 272]#? ([http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Clark-Jarvis])
|
|-
| Dec 21
|
| [http://math.mit.edu/~drew/maple_3_1640042.txt 1,640,042]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258151 Sutherland])

<strike>[http://math.mit.edu/~drew/maple_4_41862295.txt 41,862,295]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland)]</strike>

[http://math.mit.edu/~drew/maple_3_1631027.txt 1,631,027]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1630680_m3.mpl 1,630,680]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258196 xfxie])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_36000000_m4.mpl 36,000,000]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258197 xfxie]</strike>

<strike>35,127,242? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_25589558_m4.mpl 25,589,558]? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258250 xfxie])</strike>
| [http://math.mit.edu/~drew/admissible_38000_429798.txt 429,798]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_1700000_25602438.txt 25,602,438]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258124 Sutherland])

[http://math.mit.edu/~drew/admissible_36000_405528.txt 405,528]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258148 Sutherland])

<strike>825,018,354? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258154 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1631027_25533684.txt 25,533,684]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258179 Sutherland])

[http://math.mit.edu/~drew/admissible_35146_395264.txt 395,264]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comments Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_35146_395234_-190558.txt 395,234]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258194 xfxie])

[http://math.mit.edu/~drew/admissible_35146_395178.txt 395,178]? [m=2] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258198 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_25527718.txt 25,527,718]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258200 Sutherland])

<strike>685,833,596? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258203 Sutherland])</strike>

<strike>491,149,914? [m=4] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258267 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_1630680_24490758.txt 24,490,758]? [m=3] ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-258268 Sutherland])
| Optimising the explicit lower bound <math>M_k \geq \log k-O(1)</math>
|-
| Dec 22
|
| 1,628,944? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

75,000,000? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

3,400,000,000? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258411 Castryck])

5,511? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

2,114,964#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

309,954? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_74487363_m4.mpl 74,487,363]? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_1628943_m3.mpl 1,628,943]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments xfxie])
| [http://math.mit.edu/~drew/admissible_35146_395154.txt 395,154]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

[http://math.mit.edu/~drew/admissible_1630680_24490410.txt 24,490,410]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])

<strike>485,825,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258305 Sutherland])</strike>

[http://math.mit.edu/~drew/admissible_35146_395122.txt 395,122]? [m=2] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])

<strike>473,244,502? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])</strike>

1,523,781,850? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

82,575,303,678? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258419 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52130.txt 52,130]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258433 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_33661442.txt 33,661,442]?# [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258451 Sutherland])

[http://math.mit.edu/~drew/admissible_1628944_24462790.txt 24,462,790]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258452 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4316446.txt 4,316,446]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258457 Sutherland])
| A numerical precision issue was discovered in the earlier m=4 calculations
|-
| Dec 23
|
| 41,589? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_41588_m4EH.mpl 41,588]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258555 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_309661_m5EH.mpl 309,661]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258558 xfxie])

[http://math.mit.edu/~drew/maple_4_BV.txt 105,754,838]#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[https://math.mit.edu/~drew/maple_5_BV.txt 5,300,000,000]#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
| [http://math.mit.edu/~drew/admissible_1628943_24462774.txt 24,462,774]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

1,512,832,950? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258527 Sutherland])

[http://math.mit.edu/~drew/admissible_309954_4146936.txt 4,146,936]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_5511_52116.txt 52,116]? [EH] [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 Sutherland])

[http://math.mit.edu/~drew/admissible_41589_474600.txt 474,600]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258529 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474460.txt 474,460]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258569 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4143140.txt 4,143,140]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258570 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313942.txt 32,313,942]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258572 Sutherland])

2,186,561,568#? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258587 Sutherland])

[http://math.mit.edu/~drew/admissible_41588_474372.txt 474,372]? [EH] [m=4]
([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258609 Sutherland])

131,161,149,090#? [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258626 Sutherland])
|
|-
| Dec 24
|
|
| [http://math.mit.edu/~drew/admissible_41588_474320.txt 474,320]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137872.txt 4,137,872]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_1628943_24462654.txt 24,462,654]? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

1,497,901,734? [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])

[http://math.mit.edu/~drew/admissible_2114964_32313878.txt 32,313,878]#? [m=3] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 Sutherland])
|
|-
| Dec 28
|
|
| [http://math.mit.edu/~drew/admissible_41588_474296.txt 474,296]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])

[http://math.mit.edu/~drew/admissible_309661_4137854.txt 4,137,854]? [EH] [m=5] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 Sutherland])
|
|-
| Jan 2 2014
|
|
| [http://math.mit.edu/~drew/admissible_41588_474290.txt 474,290]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-260937 Sutherland])
|
|-
| Jan 6
|
| 54# ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 Nielsen])
| 270# ([http://math.mit.edu/~primegaps/tuples/admissible_54_270.txt Clark-Jarvis])
|
|-
| Jan 8
|
| 4 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| 8 [GEH] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262403 Nielsen])
| Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262511 here].
|-
| Jan 9
|
|
| [http://math.mit.edu/~drew/admissible_41588_474266.txt 474,266]? [EH] [m=4] ([http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments Sutherland])
|
|-
| Jan 28
|
|
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland])
|
|-
| Jan 29
|
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen])
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube
|-
| Feb 9
|
|
|
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here]
|-
| Feb 17
|
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen])
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis])
| Managed to get the epsilon trick to be computationally feasible for medium k
|-
| Feb 22
|
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen])
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis])
| More efficient matrix computation allows for higher degrees to be used
|-
| Mar 4
|
|
|
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed]
|-
| Apr 14
|
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen])
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis])
| A 2-week computer calculation!
|-
| Apr 17
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707]? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908]? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie])
|
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [GEH] - bound is conditional the generalized Elliott-Halberstam conjecture
# [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

Bounded gaps between primes

2014-04-16T08:55:17Z

AndrewVSutherland: /* Recent papers and notes */

Bounded gaps between primes

2013-12-23T23:02:09Z

AndrewVSutherland: /* Current records */

Bounded gaps between primes

2013-11-24T15:10:01Z

AndrewVSutherland: /* World records */

This is the home page for the Polymath8 project, which has two components:

* Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.
* Polymath8b, "Bounded intervals with many primes", is an ongoing project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard.

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>, as described in the page on [[Dickson-Hardy-Littlewood theorems]]. In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on [[distribution of primes in smooth moduli]].

In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 10 Aug 2005
|
| 6 [EH]
| 16 [EH] ([[http://arxiv.org/abs/math/0508185 Goldston-Pintz-Yildirim]])
| First bounded prime gap result (conditional on Elliott-Halberstam)
|-
| 14 May 2013
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632 ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686] ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680] ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math># ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788# ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]# ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783# ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]# ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>83\frac{1}{13}\varpi + 25\frac{5}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
12 [EH] ([http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-le/144546#144546 Maynard])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|-
| Oct 21
|
|
|
| All sections of the paper relating to the bounds obtained on Jul 27 and Aug 17 have been proofread at least twice
|-
| Oct 23
|
|
| 700#? (Maynard)
| [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248855 Announced] at a talk in Oberwolfach
|-
| Oct 24
|
| 110#? ([http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/#comment-248898 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_110_628.txt 628]#? ([http://math.mit.edu/~primegaps/ Engelsma])
| With this value of <math>k_0</math>, the value of <math>H</math> given is best possible (and similarly for smaller values of <math>k_0</math>)
|-
| Nov 19
|
| 105# ([http://arxiv.org/abs/1311.4600 Maynard])
| [http://math.mit.edu/~primegaps/tuples/admissible_105_600.txt 600]# ([http://arxiv.org/abs/1311.4600 Maynard]/[http://math.mit.edu/~primegaps/ Engelsma])
| One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam
|-
| Nov 20
|
| <strike>145*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251808 Nielsen])</strike>
<strike>13,986 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251811 Nielsen])</strike>
| <strike>[http://math.mit.edu/~primegaps/tuples/admissible_145_864.txt 864]*? ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
<strike>[http://math.mit.edu/~drew/admissible_13986_145212.txt 145,212] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251826 Sutherland])</strike>
| Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation
|-
| Nov 21
|
| 68?? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251876 Maynard])
582#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

59,451 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])

[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_508.mpl 508]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251894 xfxie])

42,392 [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251921 Nielsen])
| [http://math.mit.edu/~primegaps/tuples/admissible_68_356.txt 356]?? ([http://math.mit.edu/~primegaps Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_582_4260.txt 4,260]#*? ([http://math.mit.edu/~primegaps Engelsma])

[http://math.mit.edu/~primegaps/tuples/admissible_508_3660.txt 3,660]*? ([http://math.mit.edu/~primegaps Engelsma])
| Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta
|-
| Nov 22
|
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi1080d13_388.mpl 388]*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252229 xfxie])

448#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])

43,134 [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
| [http://math.mit.edu/~drew/admissible_59451_698288.txt 698,288] [m=2]#? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251997 Sutherland])
[https://math.mit.edu/~drew/admissible_42392_484290.txt 484,290] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252106 Sutherland])

[https://math.mit.edu/~drew/admissible_42392_484276.txt 484,276] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252138 Sutherland])

[http://math.mit.edu/~primegaps/tuples/admissible_388_2702.txt 2,702]*? ([http://math.mit.edu/~primegaps Engelsma])
| Uses the m=2 values of k_0 from Nov 21
|-
| Nov 23
|
|
| [http://math.mit.edu/~drew/admissible_43134_493528.txt 493,528] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252534 Sutherland]

[http://math.mit.edu/~drew/admissible_43134_493510.txt 493,510] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252691 Sutherland]

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484272_-211144.txt 484,272] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252819 xfxie])

[http://math.mit.edu/~drew/admissible_42392_484260.txt 484,260] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252823 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484238_-211144.txt 484,238] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252857 xfxie])

[http://math.mit.edu/~drew/admissible_43134_493458.txt 493,458] [m=2]#? [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252824 Sutherland]
|
|-
| Nov 24
|
|
| [http://math.mit.edu/~drew/admissible_42392_484234.txt 484,234] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252928 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_42392_484200_-210008.txt 484,200] [m=2]? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252951 xfxie])
|
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems
# [EH] - bound is conditional the Elliott-Halberstam conjecture
# [m=2] - bound on intervals containing three consecutive primes, rather than two
# strikethrough - values relied on a computation that has now been retracted
# boldface - the current best unconditional bound on H that we have high confidence in

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

# [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
# [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/ Bounded gaps between primes (Polymath8) – a progress report], Terence Tao, 30 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/ The quest for narrow admissible tuples], Andrew Sutherland, 2 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ The distribution of primes in doubly densely divisible moduli], Terence Tao, 7 July 2013. Inactive.
# [http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/ An improved Type I estimate], Terence Tao, 27 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ Polymath8: writing the paper], Terence Tao, 17 August 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/ Polymath8: writing the paper, II], Terence Tao, 2 September 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/ Polymath8: writing the paper, III], Terence Tao, 22 September 2013. Inactive
# [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/ Polymath8: writing the paper, IV], Terence Tao, 15 October 2013. Inactive
# [http://terrytao.wordpress.com/2013/11/17/polymath8-writing-the-first-paper-v-and-a-look-ahead/ Polymath8: Writing the first paper, V, and a look ahead], Terence Tao, 17 November 2013. Active
# [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/ Polymath8b: Bounded intervals with many primes, after Maynard], Terence Tao, 19 November 2013. Inactive
# [http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/ Polymath8b, II: Optimising the variational problem and the sieve] Terence Tao, 22 November 2013. Active

== Writeup ==

Files for the draft paper for this project may be found in [https://www.dropbox.com/sh/j2r8yia6lkzk2gv/_5Sn7mNN3T this directory]. The compiled PDF is available [https://www.dropbox.com/s/16bei7l944twojr/newgap.pdf here].

Here are the [[Polymath8 grant acknowledgments]].

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie
* [https://math.mit.edu/~primegaps/MaynardMathematicaNotebook.txt Mathematica Notebook for optimising M_k], James Maynard

=== Tuples applet ===

Here is [https://math.mit.edu/~primegaps/sieve.html?ktuple=632 a small javascript applet] that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The same applet [https://math.mit.edu/~primegaps/sieve.html can also be used to interactively create new admissible tuples]. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.

You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is

https://math.mit.edu/~primegaps/sieve.html?d=4680&r=1,1,4,3,2,8,2,14,13,8,29,31,33,28,6,49,21,47,58,35,57,44,1,55,50,57,9,91,87,45,89,50,16,19,122,114,151,66

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation/ A ternary divisor variation], Emmanuel Kowalski, 25 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/09/09/conductors-of-one-variable-transforms-of-trace-functions/ Conductors of one-variable transforms of trace functions], Emmanuel Kowalski, 9 September 2013.
* [http://gilkalai.wordpress.com/2013/09/20/polymath-8-a-success/ Polymath 8 – a Success!], Gil Kalai, 20 September 2013.
* [http://blogs.ethz.ch/kowalski/2013/10/24/james-maynard-auteur-du-theoreme-de-lannee/ James Maynard, auteur du théorème de l’année], Emmanuel Kowalski, 24 October 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://www.aimath.org/news/primegaps70m/ Zhang's Theorem on Bounded Gaps Between Primes], Dan Goldston, May? 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.
* [http://www.renyi.hu/~gharcos/gaps.pdf Lecture notes: bounded gaps between primes], Gergely Harcos, 1 Oct 2013.
* [http://math.mit.edu/~drew/PrimeGaps.pdf New bounds on gaps between primes], Andrew Sutherland, 17 Oct 2013.
* [http://www.dms.umontreal.ca/~andrew/CurrentEventsArticle.pdf Bounded gaps between primes], Andrew Granville, 29 Oct 2013.
* [http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf Primes in intervals of bounded length], Andrew Granville, 19 Nov 2013.
* [http://arxiv.org/abs/1311.4600 Small gaps between primes], James Maynard, 19 Nov 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.
* [http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html L'union fait la force des mathématiciens], Philippe Pajot, Le Monde, 24 June, 2013.
* [http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/ Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers], Amir Aczel, Discover Magazine, 10 July, 2013.
* [http://nautil.us/issue/5/fame/the-twin-prime-hero The Twin Prime Hero], Michael Segal, Nautilus, Issue 005, 2013.
* [http://news.anu.edu.au/2013/11/19/prime-time/ Prime Time], Casey Hamilton, Australian National University, 19 November 2013.
* [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ Together and Alone, Closing the Prime Gap], Erica Klarreich, Quanta, 19 November 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/11/prime/ Sudden Progress on Prime Number Problem Has Mathematicians Buzzing]".
** [http://science.slashdot.org/story/13/11/20/1256229/mathematicians-team-up-to-close-the-prime-gap Mathematicians Team Up To Close the Prime Gap], Slashdot, 20 November 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
* [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. [http://www.ams.org/mathscinet-getitem?mr=485727 MathSciNet] [http://link.springer.com/article/10.1007%2FBF01578069 Article]
* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Finding narrow admissible tuples

2013-10-13T16:24:58Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 3,912
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
|
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
|
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
|
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
|
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
|
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
|
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-10-13T13:26:59Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
|
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 3,870
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 3,828
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 3,792
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 4.052
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 4,052
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
|
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
|
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
|
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
|
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
|
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
|
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-10-13T12:04:04Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 56,238,957
| 4,694,650
| 2,372,231
| 394,096
| 290,604
| 257,404
| 253,380
| 110,188
| 70,496
| 61,726
| 52,370
| 47,585
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 603 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 4,800
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,634]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_603_4688.txt 4,688]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_603_4672.txt 4,672]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_603_4610.txt 4,610]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_603_4452.txt 4,452]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k632_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k603_4422.txt 4,422]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>\lfloor k_0 \log k_0 + k_0 \rfloor</math>
| 37,176
| 31,097
| 27,019
| 17,201
| 15,130
| 7,907
| 5,046
| 4,707
| 4,463
| 2,337
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
|
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
|
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
|
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
|
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
|
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
|
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
|
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
|
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
|
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
|
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
|
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
|
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Bounded gaps between primes

2013-10-12T16:23:33Z

AndrewVSutherland: /* Tuples applet */

This is the home page for the Polymath8 project "bounded gaps between primes".

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>, as described in the page on [[Dickson-Hardy-Littlewood theorems]]. In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on [[distribution of primes in smooth moduli]].

In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686]? ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680]? ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math>**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]**? ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783**? ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]**? ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>\frac{1080}{13}\varpi + \frac{330}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>**</nowiki> - bound does not rely on Deligne's theorems
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

# [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
# [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/ Bounded gaps between primes (Polymath8) – a progress report], Terence Tao, 30 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/ The quest for narrow admissible tuples], Andrew Sutherland, 2 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ The distribution of primes in doubly densely divisible moduli], Terence Tao, 7 July 2013. Inactive.
# [http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/ An improved Type I estimate], Terence Tao, 27 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ Polymath8: writing the paper], Terence Tao, 17 August 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/ Polymath8: writing the paper, II], Terence Tao, 2 September 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/ Polymath8: writing the paper, III], Terence Tao, 22 September 2013. Active

== Writeup ==

Files for the draft paper for this project may be found in [https://www.dropbox.com/sh/j2r8yia6lkzk2gv/_5Sn7mNN3T this directory]. The compiled PDF is available [https://www.dropbox.com/s/16bei7l944twojr/newgap.pdf here].

Here are the [[Polymath8 grant acknowledgments]].

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

=== Tuples applet ===

Here is [https://math.mit.edu/~primegaps/sieve.html?ktuple=632 a small javascript applet] that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The same applet [https://math.mit.edu/~primegaps/sieve.html can also be used to interactively create new admissible tuples]. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.

You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is

https://math.mit.edu/~primegaps/sieve.html?d=4680&r=1,1,4,3,2,8,2,14,13,8,29,31,33,28,6,49,21,47,58,35,57,44,1,55,50,57,9,91,87,45,89,50,16,19,122,114,151,66

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation/ A ternary divisor variation], Emmanuel Kowalski, 25 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/09/09/conductors-of-one-variable-transforms-of-trace-functions/ Conductors of one-variable transforms of trace functions], Emmanuel Kowalski, 9 September 2013.
* [http://gilkalai.wordpress.com/2013/09/20/polymath-8-a-success/ Polymath 8 – a Success!], Gil Kalai, 20 September 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.
* [http://www.renyi.hu/~gharcos/gaps.pdf Lecture notes: bounded gaps between primes], Gergely Harcos, 1 Oct 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.
* [http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html L'union fait la force des mathématiciens], Philippe Pajot, Le Monde, 24 June, 2013.
* [http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/ Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers], Amir Aczel, Discover Magazine, 10 July, 2013.
* [http://nautil.us/issue/5/fame/the-twin-prime-hero The Twin Prime Hero], Michael Segal, Nautilus, Issue 005, 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
* [FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. [http://www.ams.org/mathscinet-getitem?mr=719249 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa42/aa4226.pdf Article]
* [FI1985] Friedlander, John B.; Iwaniec, Henryk, Incomplete Kloosterman sums and a divisor problem. With an appendix by Bryan J. Birch and Enrico Bombieri. Ann. of Math. (2) 121 (1985), no. 2, 319–350. [http://www.ams.org/mathscinet-getitem?mr=786351 MathSciNet] [http://www.jstor.org/stable/1971175 JSTOR] [http://www.jstor.org/stable/1971176 Appendix]
* [GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. [http://arxiv.org/abs/math/0508185 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2552109 MathSciNet]
* [GR1998] Gordon, Daniel M.; Rodemich, Gene Dense admissible sets. Algorithmic number theory (Portland, OR, 1998), 216–225, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. [http://www.ams.org/mathscinet-getitem?mr=1726073 MathSciNet] [http://www.ccrwest.org/gordon/ants.pdf Article]
* [GR1980] S. W. Graham, C. J. Ringrose, Lower bounds for least quadratic nonresidues. Analytic number theory (Allerton Park, IL, 1989), 269–309, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990. [http://www.ams.org/mathscinet-getitem?mr=1084186 MathSciNet] [http://link.springer.com/content/pdf/10.1007%2F978-1-4612-3464-7_18.pdf Article]
* [HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. [http://www.ams.org/mathscinet-getitem?mr=485727 MathSciNet] [http://link.springer.com/article/10.1007%2FBF01578069 Article]
* [HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. [http://www.ams.org/mathscinet-getitem?mr=866901 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4713.pdf Article]
* [HR1973] Hensley, Douglas; Richards, Ian, On the incompatibility of two conjectures concerning primes. Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 123–127. Amer. Math. Soc., Providence, R.I., 1973. [http://www.ams.org/mathscinet-getitem?mr=340194 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/S0002-9904-1974-13434-8.pdf Article]
* [HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. [http://www.ams.org/mathscinet-getitem?mr=396440 MathSciNet] [https://eudml.org/doc/205282 Article]
* [MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. [http://arxiv.org/abs/math/0602599 arXiv] [http://www.ams.org/mathscinet-getitem?mr=2414788 MathSciNet] [http://blms.oxfordjournals.org/content/40/2/298 Article]
* [MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. [http://www.ams.org/mathscinet-getitem?mr=374060 MathSciNet] [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=6718308 Article]
* [M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. [http://www.ams.org/mathscinet-getitem?mr=466048 MathSciNet] [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183540922 Article]
* [R1974] Richards, Ian On the incompatibility of two conjectures concerning primes; a discussion of the use of computers in attacking a theoretical problem. Bull. Amer. Math. Soc. 80 (1974), 419–438. [http://www.ams.org/mathscinet-getitem?mr=337832 MathSciNet] [http://www.ams.org/journals/bull/1974-80-03/S0002-9904-1974-13434-8/home.html Article]
* [S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. [http://www.ams.org/mathscinet-getitem?mr=130203 MathSciNet] [http://matwbn.icm.edu.pl/ksiazki/aa/aa7/aa711.pdf Article]
* [S2007] K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım. Bull. Amer. Math. Soc. (N.S.) 44 (2007), no. 1, 1–18. [http://www.ams.org/mathscinet-getitem?mr=2265008 MathSciNet] [http://www.ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/ Article] [http://arxiv.org/abs/math/0605696 arXiv]

Bounded gaps between primes

2013-10-12T16:21:43Z

AndrewVSutherland: /* Polymath threads */

This is the home page for the Polymath8 project "bounded gaps between primes".

== World records ==

* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in <math>k_0</math> lead to improvements in <math>H</math>. (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].)
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture]. Would like to be as large as possible. Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>, as described in the page on [[Dickson-Hardy-Littlewood theorems]]. In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on [[distribution of primes in smooth moduli]].

In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.

{| border=1
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
|-
| 14 May
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
| All subsequent work is based on Zhang's breakthrough paper.
|-
| 21 May
|
|
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
| Optimises Zhang's condition <math>\pi(H)-\pi(k_0) > k_0</math>; [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23439 can be reduced by 1] by parity considerations
|-
| 28 May
|
|
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
|-
| 30 May
|
|
| 59,470,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/ Morrison])
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])

59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])

57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
|-
| 31 May
|
| 2,947,442 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
| 48,112,378 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23460 Morrison])
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])

42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
|-
| 1 Jun
|
|
| 42,342,924 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 Tao])
| Tiny improvement using the parity of <math>k_0</math>
|-
| 2 Jun
|
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
|-
| 3 Jun
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
|-
| 4 Jun
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
|
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
| Uses asymmetric version of the Hensley-Richards tuples
|-
| 5 Jun
|
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])

397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])

4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])

388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])

388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])

388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])

[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])

387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])

387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])

| <math>k_0</math> bound uses the optimal Bessel function cutoff. Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve

|-
| 6 Jun
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])

<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])

387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])

[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])

[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])

<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])

| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
|-
| 7 Jun
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])

<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])

<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])

26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])

<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])

<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])

<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])

<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])

<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])

<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)

<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)

<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])

<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])

387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
|-
| Jun 8
|
|
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])

285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])

[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox]. New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
|-
| Jun 9
|
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])

[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])

[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
|-
| Jun 10
|
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])

[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])

[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
|-
| Jun 11
|
|
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])

[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
|-
| Jun 12
|
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])

[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
|-
| Jun 13
|
|
|

[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])

[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
|
|-
| Jun 14
|
|
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
|
|-
| Jun 15
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])

6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
|-
| Jun 16
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])

<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])

<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>

<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>

<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
|-
| Jun 18
|
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])

5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])

5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])

5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])

| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])

[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)

58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])

[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])

53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])

53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
|-
| Jun 19
|
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])

5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])

5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])

[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])

[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])

[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])

53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])

[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
|-
| Jun 20
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])

<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
|
|
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
|-
| Jun 21
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])

1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])

[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
|-
| Jun 22
|
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
|-
| Jun 23
|
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
|-
| Jun 24
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])

<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])

<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>

<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
|-
| Jun 25
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])

<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>

1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])

<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>

[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])

| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
|-
| Jun 26
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])

<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums. Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
|-
| Jun 27
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums. [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
|-
| Jul 1
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
|
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])

<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
|
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])

<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
|-
| Jul 5
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
|
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
|
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
|
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
|-
| Jul 10
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
|
|
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
|-
| Jul 19
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
|
|
| A more detailed computation of the Jul 10 refinement
|-
| Jul 20
|
|
|
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
|-
| Jul 27
|
| 633? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
632? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686]? ([http://math.mit.edu/~primegaps/ Engelsma])
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680]? ([http://math.mit.edu/~primegaps/ Engelsma])
|
|-
| Jul 30
| <math>168\varpi + 48\delta < 1</math>**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| 1,788**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]**? ([http://math.mit.edu/~primegaps/ Sutherland])
| Bound obtained without using Deligne's theorems.
|-
| Aug 17
|
| 1,783**? ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]**? ([http://math.mit.edu/~primegaps/ Sutherland])
|
|-
| Oct 3
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
| Found an additional variable to apply van der Corput to
|-
| Oct 11
| <math>\frac{1080}{13}\varpi + \frac{330}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
|}

Legend:
# ? - unconfirmed or conditional
# ?? - theoretical limit of an analysis, rather than a claimed record
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
# <nowiki>**</nowiki> - bound does not rely on Deligne's theorems
# strikethrough - values relied on a computation that has now been retracted

See also the article on ''[[Finding narrow admissible tuples]]'' for benchmark values of <math>H</math> for various key values of <math>k_0</math>.

== Polymath threads ==

# [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart], Scott Morrison, 30 May 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang], Terence Tao, 3 June 2013. Inactive
# [http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/ Polymath proposal: bounded gaps between primes], Terence Tao, 4 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/ Online reading seminar for Zhang’s “bounded gaps between primes”], Terence Tao, 4 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/ More narrow admissible sets], Scott Morrison, 5 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/ The elementary Selberg sieve and bounded prime gaps], Terence Tao, 8 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/10/a-combinatorial-subset-sum-problem-associated-with-bounded-prime-gaps/ A combinatorial subset sum problem associated with bounded prime gaps], Terence Tao, 10 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Further analysis of the truncated GPY sieve], Terence Tao, 11 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/ Estimation of the Type I and Type II sums], Terence Tao, 12 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/ Estimation of the Type III sums], Terence Tao, 14 June 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/ A truncated elementary Selberg sieve of Pintz], Terence Tao, 18 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/ Bounding short exponential sums on smooth moduli via Weyl differencing], Terence Tao, 22 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/ The distribution of primes in densely divisible moduli], Terence Tao, 23 June, 2013. Inactive
# [http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/ Bounded gaps between primes (Polymath8) – a progress report], Terence Tao, 30 June 2013. Inactive
# [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/ The quest for narrow admissible tuples], Andrew Sutherland, 2 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/ The distribution of primes in doubly densely divisible moduli], Terence Tao, 7 July 2013. Inactive.
# [http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/ An improved Type I estimate], Terence Tao, 27 July 2013. Inactive
# [http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ Polymath8: writing the paper], Terence Tao, 17 August 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/ Polymath8: writing the paper, II], Terence Tao, 2 September 2013. Inactive
# [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/ Polymath8: writing the paper, III], Terence Tao, 22 September 2013. Active

== Writeup ==

Files for the draft paper for this project may be found in [https://www.dropbox.com/sh/j2r8yia6lkzk2gv/_5Sn7mNN3T this directory]. The compiled PDF is available [https://www.dropbox.com/s/16bei7l944twojr/newgap.pdf here].

Here are the [[Polymath8 grant acknowledgments]].

== Code and data ==

* [https://github.com/semorrison/polymath8 Hensely-Richards sequences], Scott Morrison
** [http://tqft.net/misc/finding%20k_0.nb A mathematica notebook for finding k_0], Scott Morrison
** [https://github.com/avi-levy/dhl python implementation], Avi Levy
* [http://www.opertech.com/primes/k-tuples.html k-tuple pattern data], Thomas J Engelsma
** [https://perswww.kuleuven.be/~u0040935/k0graph.png A graph of this data]
** [http://www.opertech.com/primes/webdata/ Tuples giving this data]
* [https://github.com/vit-tucek/admissible_sets Sifted sequences], Vit Tucek
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie

=== Tuples applet ===

Here is [https://math.mit.edu/~primegaps/sieve.html?ktuple=632 a small javascript applet] that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The same applet [https://math.mit.edu/~primegaps/sieve.html can also be used to interactively create new admissible tuples]. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.

You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is

https://math.mit.edu/~primegaps/sieve.html?d=4680&r=1,1,4,3,2,8,2,14,13,8,29,31,33,28,6,49,21,47,58,35,57,44,1,55,50,57,9,91,87,45,89,50,16,19,122,114,151,66

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and I have only tested it on the current versions of Chrome and Firefox (it should work in most browsers but I make no guarantees). Also, the MIT math web server has recently developed the habit of occasionally throwing 403 errors when you try to load a page — if this happens just hit reload.

== Errata ==

Page numbers refer to the file linked to for the relevant paper.

# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
## Page 35: In the display after (10.5), the subscript on <math>{\mathcal J}_i</math> should be deleted.
## Page 36: In the third display, a factor of <math>\tau(q_0r)^{O(1)}</math> may be needed on the right-hand side (but is ultimately harmless).
## Page 38: In the display after (10.14), <math>\xi(r,a;q_1,b_1;q_2,b_2;n,k)</math> should be <math>\xi(r,a;k;q_1,b_1;q_2,b_2;n)</math>.
## Page 42: In (12.3), <math>B</math> should probably be 2.
## Page 47: In the third display after (13.13), the condition <math>l \in {\mathcal I}_i(h)</math> should be <math>l \in {\mathcal I}_i(sh)</math>.
## Page 49: In the top line, a comma in <math>(h_1,h_2;,n_1,n_2)</math> should be deleted.
## Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
## Page 54: Three displays before (14.17), <math>\bar{r_2}(m_1+m_2)q</math> should be <math>\bar{r_2}(m_1+m_2)/q</math>.
# Errata for Motohashi-Pintz's "[http://arxiv.org/pdf/math/0602599v1.pdf A smoothed GPY sieve]", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
## Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of <math>(\log \frac{R}{|D|})^{2\ell+1}</math>, (4.15) contains instead a factor of <math>(\log \frac{R/w}{|K|})^{2\ell+1}</math> which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of <math>\exp(-k\omega/3)</math> in (4.15) does not seem to be available for estimating the second sum in (5.14).
# Errata for Pintz's "[http://arxiv.org/pdf/1306.1497v1.pdf A note on bounded gaps between primes]", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
## Page 7: In (2.39), the exponent of <math>3a/2</math> should instead be <math>-5a/2</math> (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
## Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

== Other relevant blog posts ==

* [http://terrytao.wordpress.com/2008/11/19/marker-lecture-iii-small-gaps-between-primes/ Marker lecture III: “Small gaps between primes”], Terence Tao, 19 Nov 2008.
* [http://blogs.ethz.ch/kowalski/2009/01/22/the-goldston-pintz-yildirim-result-and-how-far-do-we-have-to-walk-to-twin-primes/ The Goldston-Pintz-Yildirim result, and how far do we have to walk to twin primes ?], Emmanuel Kowalski, 22 Jan 2009.
* [http://www.math.columbia.edu/~woit/wordpress/?p=5865 Number Theory News], Peter Woit, 12 May 2013.
* [http://golem.ph.utexas.edu/category/2013/05/bounded_gaps_between_primes.html Bounded Gaps Between Primes], Emily Riehl, 14 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ Bounded gaps between primes!], Emmanuel Kowalski, 21 May 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/04/bounded-gaps-between-primes-some-grittier-details/ Bounded gaps between primes: some grittier details], Emmanuel Kowalski, 4 June 2013.
** [http://www.math.ethz.ch/~kowalski/zhang-notes.pdf The slides from the talk mentioned in that post]
* [http://aperiodical.com/2013/06/bound-on-prime-gaps-bound-decreasing-by-leaps-and-bounds/ Bound on prime gaps bound decreasing by leaps and bounds], Christian Perfect, 8 June 2013.
* [http://blogs.ams.org/blogonmathblogs/2013/06/14/narrowing-the-gap/ Narrowing the Gap], Brie Finegold, AMS Blog on Math Blogs, 14 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/22/bounded-gaps-between-primes-the-dawn-of-some-enlightenment/ Bounded gaps between primes: the dawn of (some) enlightenment], Emmanuel Kowalski, 22 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation/ A ternary divisor variation], Emmanuel Kowalski, 25 June 2013.
* [http://blogs.ethz.ch/kowalski/2013/09/09/conductors-of-one-variable-transforms-of-trace-functions/ Conductors of one-variable transforms of trace functions], Emmanuel Kowalski, 9 September 2013.
* [http://gilkalai.wordpress.com/2013/09/20/polymath-8-a-success/ Polymath 8 – a Success!], Gil Kalai, 20 September 2013.

== MathOverflow ==

* [http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture], 20 May 2013.
* [http://mathoverflow.net/questions/131825/a-technical-question-related-to-zhangs-result-of-bounded-prime-gaps A technical question related to Zhang’s result of bounded prime gaps], 25 May 2013.
* [http://mathoverflow.net/questions/132452/how-does-yitang-zhang-use-cauchys-inequality-and-theorem-2-to-obtain-the-error How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the <math>S_2</math> sum], 31 May 2013.
* [http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed Tightening Zhang’s bound], 3 June 2013.
** [http://meta.mathoverflow.net/discussion/1605/tightening-zhangs-bound/ Metathread for this post]
* [http://mathoverflow.net/questions/132731/does-zhangs-theorem-generalize-to-3-or-more-primes-in-an-interval-of-fixed-len Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?], 3 June 2013.

== Wikipedia and other references ==

* [http://en.wikipedia.org/wiki/Bessel_function Bessel function]
* [http://en.wikipedia.org/wiki/Bombieri-Vinogradov_theorem Bombieri-Vinogradov theorem]
* [http://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh_theorem Brun-Titchmarsh theorem]
* [http://www.encyclopediaofmath.org/index.php/Dispersion_method Dispersion method]
* [http://en.wikipedia.org/wiki/Prime_gap Prime gap]
* [http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture Second Hardy-Littlewood conjecture]
* [http://en.wikipedia.org/wiki/Twin_prime_conjecture Twin prime conjecture]

== Recent papers and notes ==

* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
* [http://arxiv.org/abs/1306.0948 Bounded length intervals containing two primes and an almost-prime II], James Maynard, 5 June 2013.
* [http://arxiv.org/abs/1306.1497 A note on bounded gaps between primes], Janos Pintz, 6 June 2013.
* [https://sites.google.com/site/avishaytal/files/Primes.pdf Lower Bounds for Admissible k-tuples], Avishay Tal, 15 June 2013.
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.
* [http://www.renyi.hu/~gharcos/gaps.pdf Lecture notes: bounded gaps between primes], Gergely Harcos, 1 Oct 2013.

== Media ==

* [http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989 First proof that infinitely many prime numbers come in pairs], Maggie McKee, Nature, 14 May 2013.
* [http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html Proof that an infinite number of primes are paired], Lisa Grossman, New Scientist, 14 May 2013.
* [https://www.simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/ Unheralded Mathematician Bridges the Prime Gap], Erica Klarreich, Simons science news, 20 May 2013.
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/05/twin-primes/ Unknown Mathematician Proves Elusive Property of Prime Numbers]".
* [http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.html The Beauty of Bounded Gaps], Jordan Ellenberg, Slate, 22 May 2013.
* [http://www.newscientist.com/article/dn23644 Game of proofs boosts prime pair result by millions], Jacob Aron, New Scientist, 4 June 2013.
* [http://www.lemonde.fr/sciences/article/2013/06/24/l-union-fait-la-force-des-mathematiciens_3435624_1650684.html L'union fait la force des mathématiciens], Philippe Pajot, Le Monde, 24 June, 2013.
* [http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/ Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers], Amir Aczel, Discover Magazine, 10 July, 2013.
* [http://nautil.us/issue/5/fame/the-twin-prime-hero The Twin Prime Hero], Michael Segal, Nautilus, Issue 005, 2013.

== Bibliography ==

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

* [BFI1986] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. Acta Math. 156 (1986), no. 3-4, 203–251. [http://www.ams.org/mathscinet-getitem?mr=834613 MathSciNet] [http://link.springer.com/article/10.1007%2FBF02399204 Article]
* [BFI1987] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. II. Math. Ann. 277 (1987), no. 3, 361–393. [http://www.ams.org/mathscinet-getitem?mr=891581 MathSciNet] [https://eudml.org/doc/164255 Article]
* [BFI1989] Bombieri, E.; Friedlander, J. B.; Iwaniec, H. Primes in arithmetic progressions to large moduli. III. J. Amer. Math. Soc. 2 (1989), no. 2, 215–224. [http://www.ams.org/mathscinet-getitem?mr=976723 MathSciNet] [http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Article]
* [B1995] Jörg Brüdern, Einführung in die analytische Zahlentheorie, Springer Verlag 1995
* [CJ2001] Clark, David A.; Jarvis, Norman C.; Dense admissible sequences. Math. Comp. 70 (2001), no. 236, 1713–1718 [http://www.ams.org/mathscinet-getitem?mr=1836929 MathSciNet] [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/home.html Article]
* [FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). [http://www.ams.org/mathscinet-getitem?mr=610700 MathSciNet] [http://www.math.ethz.ch/~kowalski/fouvry-iwaniec-on-a-theorem.pdf Article]
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Finding narrow admissible tuples

2013-09-11T10:11:36Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 35,926,668
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-09T09:06:09Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
| 304,704
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-08T17:58:58Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
| 3,379,776
| 1,739,850
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-08T17:54:43Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 24,226,450
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-06T01:57:40Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,344
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
|
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-05T08:24:20Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
| 226,104
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,356
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
|
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-05T08:22:18Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
|
| 200,852
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,356
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
|
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-04T21:53:32Z

AndrewVSutherland: Removed incorrect partition bounds with p_exh=11

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
|
|
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,356
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
|
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.

Finding narrow admissible tuples

2013-09-04T21:19:58Z

AndrewVSutherland: /* Benchmarks */

For any natural number <math>k_0</math>, an ''admissible <math>k_0</math>-tuple'' is a finite set <math>{\mathcal H}</math> of integers of cardinality <math>k_0</math> which avoids at least one residue class modulo <math>p</math> for each prime <math>p</math>. (Note that one only needs to check those primes <math>p</math> of size at most <math>k_0</math>, so this is a finitely checkable condition.) Let <math>H(k_0)</math> denote the minimal diameter <math>\max {\mathcal H} - \min {\mathcal H}</math> of an admissible <math>k_0</math>-tuple. As part of the [[Bounded gaps between primes|Polymath8]] project, we would like to find as good an upper bound on <math>H(k_0)</math> as possible for given values of <math>k_0</math>. To a lesser extent, we would also be interested in lower bounds on this quantity. There is some scattered numerical evidence that the optimal value of H is roughly of size <math>k_0 \log k_0 + k_0</math> for <math>k_0</math> in the range of interest.

== Upper bounds ==

Upper bounds are primarily constructed through various "sieves" that delete one residue class modulo <math>p</math> from an interval for a lot of primes <math>p</math>. Examples of sieves, in roughly increasing order of efficiency, are listed below.

=== Zhang sieve ===

The Zhang sieve uses the tuple

:<math>{\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\}</math>

where <math>m</math> is minimized subject to staying admissible. Any <math>m</math> with <math>p_{m+1} > k_0</math> yields an admissible tuple; in particular, one can just take <math>{\mathcal H}</math> to be the first <math>k_0</math> primes past <math>k_0</math>, but this is not optimal. Applying the prime number theorem then gives the upper bound <math>H \leq (1+o(1)) k_0\log k_0</math>.

=== Hensley-Richards sieve ===

The Hensley-Richards sieve [HR1973], [HR1973b], [R1974] uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1}\}</math>

where m is optimised to minimize the diameter while staying admissible.

=== Asymmetric Hensley-Richards sieve ===

The asymmetric Hensley-Richard sieve uses the tuple

:<math>{\mathcal H} = \{-p_{m+\lfloor k_0/2\rfloor - 1-i}, \ldots, -p_{m+1}, -1, +1, p_{m+1},\ldots, p_{m+\lfloor k_0/2+1/2\rfloor-1+i}\}</math>

where <math>i</math> is an integer and <math>i,m</math> are optimised to minimize the diameter while staying admissible.

=== Schinzel sieve ===
Given <math>0<y<z<x</math>, the Schinzel sieve (discussed in [S1961], [HR1973], [GR1998], [CJ2001]) sieves the interval <math>[1,x]</math> by <math>1 \bmod p</math> for primes <math>p \le y</math> and by <math>0\bmod p</math> for primes <math>y . Provided that <math>z</math> is large enough (<math>z=k_0</math> clearly suffices), the first <math>k_0</math> survivors form an admissible <math>k_0</math>-tuple (but not necessarily the narrowest one in the interval).
The case <math>y=1</math> corresponds to a sieve of Eratosthenes; if one minimizes <math>z</math> and takes the first <math>k_0</math> survivors greater than 1, this yields the same admissible <math>k_0</math> tuple as Zhang, with the minimal possible value of <math>m</math>.

=== Shifted Schinzel sieve ===
As a generalization of the Schinzel sieve, one may instead sieve shifted intervals <math>[s,s+x]</math>. This is effectively equivalent to sieving the interval <math>[0,x]</math> of the residue classes <math>-s\ \bmod\ p</math> for primes <math>p\le y</math> and <math> 1-s\ \bmod\ p</math> for primes <math>y<p\le z</math>.

=== Greedy sieve ===
Within a given interval, one sieves a single residue class <math>a \bmod p</math> for increasing primes <math>p=2,3,5,\ldots</math>, with <math>a</math> chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize <math>a\in[0,p-1]</math>, maximize <math>a\in [0,p-1]</math>, minimize <math>|a-\lfloor p/2\rfloor|</math>, or randomly. If not all residue classes modulo <math>p</math> are occupied by survivors, then <math>a</math> will be chosen so that no survivors are sieved.
This necessarily occurs once <math>p</math> exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest <math>k_0</math>-tuple <math>{\mathcal H}</math> among the survivors (if there are fewer than <math>k_0</math> survivors, retry with a wider interval).

=== Greedy-greedy sieve ===
Heuristically, the performance of the greedy sieve is significantly improved by starting with a shifted Schinzel sieve on <math>[s,\ s+x]</math> using <math>y = 2</math> and <math>z = \sqrt{x}</math> and then continuing in a greedy fashion, as [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart#comment-23566 proposed by Sutherland]. One first optimizes the shift value <math>s</math> over some larger interval (e.g. <math>[-k_0\log\ k_0,\ k_0\log\ k_0]</math>) and then continues the sieving over primes <math>p > z</math> greedily choosing the best residue class for each prime according to a chosen tie-breaking rule (in Sutherland's original implementation, ties are broken downward in <math>[0,\ p-1]</math>).

=== Seeded greedy sieve ===
Given an initial sequence <math>{\mathcal S}</math> that is known to contain an admissible <math>k_0</math>-tuple, one can apply greedy sieving to the minimal interval containing <math>{\mathcal S}</math> until an admissible sequence of survivors remains, and then choose the narrowest <math>k_0</math>=tuple it contains. The sieving methods above can be viewed as the special case where <math>{\mathcal S}</math> is the set of integers in some interval. The main difference is that the choice of <math>{\mathcal S}</math> affects when ties occur and how they are broken with greedy sieving.
One approach is to take <math>{\mathcal S}</math> to be the union of two <math>k_0</math>-tuples that lie in roughly the same interval (see Iterated merging) below.

=== Iterated merging ===
Given an admissible <math>k_0</math>-tuple <math>\mathcal{H}_1</math>, one can attempt to improve it using an iterated merging approach [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 suggested by Castryck]. One first uses a greedy (or greedy-Schinzel) sieve to construct an admissible <math>k_0</math>-tuple <math>\mathcal{H}_2</math> in roughly the same interval as <math>\mathcal{H}_1</math>, then performs a randomized greedy sieve using the seed set <math>\mathcal{S} = \mathcal{H}_1 \cup \mathcal{H}_2</math> to obtain an admissible <math>k_0</math>-tuple <math>\mathcal{H}_3</math>. If <math>\mathcal{H}_3</math> is narrower than <math>\mathcal{H}_2</math>, replace <math>\mathcal{H}_2</math> with <math>\mathcal{H}_3</math>, otherwise try again with a new <math>\mathcal{H}_3</math>.
Eventually the diameter of <math>\mathcal{H}_2</math> will become less than or equal to that of <math>\mathcal{H}_1</math>.
As long as <math>\mathcal{H}_1\ne \mathcal{H}_2</math>, one can continue to attempt to improve <math>\mathcal{H}_2</math>, but in practice one stops after some number of retries.

As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23733 described by Sutherland], one can then replace <math>\mathcal{H}_1</math> with <math>\mathcal{H}_2</math> and begin the process anew, yielding a randomized algorithm that can be run indefinitely. Key parameters to this algorithm are the choice of the interval used when constructing <math>\mathcal{H}_2</math>, which is typically made wider than the minimal interval containing <math>\mathcal{H}_1</math> by a small factor <math>\delta</math> on each side (Sutherland suggests <math>\delta = 0.0025</math>), and the number of failed attempts allowed while attempting to impove <math>\mathcal{H}_2</math>.

Eventually this process will tend to converge to particular <math>\mathcal{H}_1</math> that it cannot improve (or more generally, a set of similar <math>\mathcal{H}_1</math>'s with the same diameter). Interleaving iterated merging with the local optimizations described below often allows the algorithm to make further progress.

----

Iterated merging can be viewed as a form of simulated annealing. The set <math>\mathcal{S}</math> initially contains at least two admissible <math>k_0</math>-tuples (typically many more), and as the algorithm proceeds the set <math>\mathcal{S}</math> converges toward <math>\mathcal{H}_1</math> and the number of admissible <math>k_0</math>-tuples it contains declines. One can regard the cardinality of the difference between <math>\mathcal{S}</math> and <math>\mathcal{H}_1</math> as a measure of the "temperature" of a gradually cooling system, since the number of choices available to the algorithm declines as this cardinality is reduced (more precisely, one may consider the entropy of the possible sequence of tie-breaking choices available for a given <math>\mathcal{S}</math>).

=== Local optimizations ===

Let <math>\mathcal H = \{h_1,\ldots, h_{k_0}\}</math> be an admissible <math>k_0</math>-tuple with endpoints <math>h_1</math> and <math>h_{k_0}</math>, and let <math>\mathcal I</math> be the interval <math>[h_1,h_{k_0}]</math>. If there exists an integer <math>h\in\mathcal I</math> such that removing one of <math>\mathcal H</math>'s endpoints and inserting <math>h</math> yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, then call <math>\mathcal H</math> ''contractible'', and if not, say that <math>\mathcal H</math> non-contractible. Note that <math>\mathcal H'</math> necessarily has smaller diameter than <math>\mathcal H</math>. Any of the sieving methods described above may produce admissible <math>k_0</math>-tuples that are contractible, so it is worth testing for contractibility as a post-processing step after sieving and replacing <math>\mathcal H</math> by <math>\mathcal H'</math> if this test succeeds.

We can also ''shift'' <math>\mathcal H </math> to the left by removing its right end point <math>h_{k_0}</math> and replacing it with the greatest integer <math>h_0 < h_1</math> that yields an admissible <math>k_0</math>-tuple <math>\mathcal H'</math>, and we can similarly shift <math>\mathcal H</math> to the right. The diameter of <math>\mathcal H'</math> need not be less than <math>\mathcal H</math>, but if it is, it provides a useful replacement. More generally, by shifting <math>\mathcal H</math> repeatedly we can produce a sequence of admissible <math>k_0</math>-tuples that lie successively further to the left or right. In general the diameter of these tuples may grow as we do so, but it will also occasionally decline, and we may be able to find a shifted <math>\mathcal H'</math> with smaller diameter than <math>\mathcal H</math>.

----

A more sophisticated local optimization involves a process of ``adjustment" [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23766 proposed by Savitt].
Let <math>\mathcal H </math> be an admissible <math>k_0</math>-tuple.
For a prime <math>p</math> and an integer <math>a</math>, let <math>[a;p]</math> denote the residue class <math>a\bmod p</math>, i.e. the set of integers <math>\{ x : x = a \bmod p\}</math>.
Call <math>[a;p]</math> occupied if it contains an element of <math>\mathcal H </math>.

Suppose that <math>[a;p]</math> and <math>[b;q]</math> are occupied residue classes, for some distinct primes <math>p</math> and <math>q</math>, and that <math>[a';p]</math> and <math>[b';q]</math> are unoccupied.
Let <math>\mathcal U</math> be the intersection of <math>\mathcal H</math> with <math>[a;p] \cup [b;q]</math>, and let <math>\mathcal V</math> be a subset of the integers that lie in the intersection of the interval <math>I</math> containing <math>H</math> and the set <math>[a';p] \cup [b';q]</math> such that
the set <math>\mathcal H' </math> formed by removing the elements of <math>\mathcal U</math> from <math>\mathcal H </math> and adding the elements of <math>\mathcal V </math> is admissible.
A necessary (and often sufficient) condition for and integer <math>v</math> to lie in <math>\mathcal V</math> is that <math>v</math> must not lie in a residue class <math>[c;r]</math> that is the unique unoccupied residue class modulo <math>r</math> for any prime <math>r</math> other than <math>p</math> or <math>q</math>.

The admissible set <math>\mathcal H' </math> lies in the interval <math>\mathcal I</math> containing <math>\mathcal H</math>, so its diameter is no greater than that of <math>\mathcal H</math>, however its cardinality may differ. If it happens that <math>\mathcal H' </math> contains more elements than <math>\mathcal H </math>, then by eliminating points at either end of <math>\mathcal H' </math> we obtain an admissible <math>k_0</math>-tuple that is narrower than <math>\mathcal H</math> and may ``adjust" <math>\mathcal H </math> by replacing it with <math>\mathcal H' </math>.
The process of adjustment can often be applied repeatedly, yielding a sequence of successively narrower admissible <math>k_0</math>-tuples.

=== Further refinements ===

== Lower bounds ==

There is a substantial amount of literature on bounding the quantity <math>\pi(x+y)-\pi(x)</math>, the number of primes in a shifted interval <math>[x+1,x+y]</math>, where <math>x,y</math> are natural numbers. As a general rule, whenever a bound of the form

: <math> \pi(x+y) - \pi(x) \leq F(y) </math> (*)

is established for some function <math>F(y)</math> of <math>y</math>, the method of proof also gives a bound of the form

: <math> k_0 \leq F( H(k_0)+1 ).</math> (**)

Indeed, if one assumes the prime tuples conjecture, any admissible <math>k_0</math>-tuple of diameter <math>H</math> can be translated into an interval of the form <math>[x+1,x+H+1]</math> for some <math>x</math>. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for <math>x>y</math>, the set of primes between <math>x+1</math> and <math>x+y</math> is admissible, so the method of proof of (*) invariably also gives (**) as well.

Examples of lower bounds are as follows;

=== Brun-Titchmarsh inequality ===

The Brun-Titchmarsh theorem gives

: <math> \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y}</math>

which then gives the lower bound

: <math> H(k_0) \geq (\frac{1}{2}-o(1)) k_0 \log k_0</math>.

Montgomery and Vaughan deleted the o(1) error from the Brun-Titchmarsh theorem [MV1973, Corollary 2], giving the more precise inequality

: <math> k_0 \leq 2 \frac{H(k_0)+1}{\log (H(k_0)+1)}.</math>

=== First Montgomery-Vaughan large sieve inequality ===

The first Montgomery-Vaughan large sieve inequality [MV1973, Theorem 1] gives

: <math> k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0)+1 + Q^2</math>

for any <math>Q > 1</math>, which is a parameter that one can optimise over (the optimal value is comparable to <math>H(k_0)^{1/2}</math>).

=== Second Montgomery-Vaughan large sieve inequality ===

The second Montgomery-Vaughan large sieve inequality [MV1973, Corollary 1] gives

: <math> k_0 \leq (\sum_{q \leq z} (H(k_0)+1+cqz)^{-1} \mu(q)^2 \prod_{p|q} \frac{1}{p-1})^{-1}</math>

for any <math>z > 1</math>, which is a parameter similar to <math>Q</math> in the previous inequality, and <math>c</math> is an absolute constant. In the original paper of Montgomery and Vaughan, <math>c</math> was taken to be <math>3/2</math>; this was then reduced to <math>\sqrt{22}/\pi</math> [B1995, p.162] and then to <math>3.2/\pi</math> [M1978]. It is conjectured that <math>c</math> can be taken to in fact be <math>1</math>.

== Benchmarks ==

Efforts to fill in the blank fields in this table are very welcome.

{| border=1
|-
!<math>k_0</math>!! 3,500,000 !! 341,640 !! 181,000 !! 34,429 !! 26,024 !! 23,283 !! 22,949 !! 10,719 !! 7,140 !! 6,329 !! 5,453 !! 5,000
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| [http://arxiv.org/abs/1305.6369 59,874,594]
| 5,005,362
| 2,530,338
| 420,878
| 310,134
| 275,082
| 270,698
| 117,714
| 75,222
| 65,924
| 55,892
| 50,840
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_3500000_59093364.txt 59,093,364]
| [http://math.mit.edu/~drew/admissible_341640_4923060.txt 4,923,060]
| [http://math.mit.edu/~drew/admissible_181000_2486370.txt 2,486,370]
| [http://math.mit.edu/~drew/admissible_34429_411946.txt 411,946]
| [http://math.mit.edu/~drew/admissible_26024_303558.txt 303,558]
| [http://math.mit.edu/~drew/admissible_23283_268544.txt 268,544]
| [http://math.mit.edu/~drew/admissible_22949_264460.txt 264,460]
| [http://math.mit.edu/~drew/admissible_10719_114814.txt 114,814]
| [http://math.mit.edu/~drew/admissible_7140_73448.txt 73,448]
| [http://math.mit.edu/~drew/admissible_6329_64182.txt 64,182]
| [http://math.mit.edu/~drew/admissible_5453_54488.txt 54,488]
| [http://math.mit.edu/~drew/admissible_5000_49578.txt 49,578]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_3500000_57554086.txt 57,554,086]
| [http://math.mit.edu/~drew/admissible_341640_4802222.txt 4,802,222]
| [http://math.mit.edu/~drew/admissible_181000_2422558.txt 2,422,558]
| [http://math.mit.edu/~drew/admissible_34429_402790.txt 402,790]
| [http://math.mit.edu/~drew/admissible_26024_297454.txt 297,454]
| [http://math.mit.edu/~drew/admissible_23283_262794.txt 262,794]
| [http://math.mit.edu/~drew/admissible_22949_258780.txt 258,780]
| [http://math.mit.edu/~drew/admissible_10719_112868.txt 112,868]
| [http://math.mit.edu/~drew/admissible_7140_72538.txt 72,538]
| [http://math.mit.edu/~drew/admissible_6329_63708.txt 63,708]
| [http://math.mit.edu/~drew/admissible_5453_53654.txt 53,654]
| [http://math.mit.edu/~drew/admissible_5000_48634.txt 48,634]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_3500000_57480832.txt 57,480,832]
| [http://math.mit.edu/~drew/admissible_341640_4788240.txt 4,788,240]
| [http://math.mit.edu/~drew/admissible_181000_2418054.txt 2,418,054]
| [http://math.mit.edu/~drew/admissible_34429_401700.txt 401,700]
| [http://math.mit.edu/~drew/admissible_26024_296154.txt 296,154]
| [http://math.mit.edu/~drew/admissible_23283_262286.txt 262,286]
| [http://math.mit.edu/~drew/admissible_22949_258302.txt 258,302]
| [http://math.mit.edu/~drew/admissible_10719_112562.txt 112,562]
| [http://math.mit.edu/~drew/admissible_7140_72062.txt 72,062]
| [http://math.mit.edu/~drew/admissible_6329_62900.txt 62,900]
| [http://math.mit.edu/~drew/admissible_5453_53278.txt 53,278]
| [http://math.mit.edu/~drew/admissible_5000_48484.txt 48,484]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_3500000_56789070.txt 56,789,070]
| [http://math.mit.edu/~drew/admissible_341640_4740846.txt 4,740,846]
| [http://math.mit.edu/~drew/admissible_181000_2396594.txt 2,396,594]
| [http://math.mit.edu/~drew/admissible_34429_399248.txt 399,248]
| [http://math.mit.edu/~drew/admissible_26024_294810.txt 294,810]
| [http://math.mit.edu/~drew/admissible_23283_260714.txt 260,714]
| [http://math.mit.edu/~drew/admissible_22949_256702.txt 256,702]
| [http://math.mit.edu/~drew/admissible_10719_112200.txt 112,200]
| [http://math.mit.edu/~drew/admissible_7140_71930.txt 71,930]
| [http://math.mit.edu/~drew/admissible_6329_62892.txt 62,892]
| [http://math.mit.edu/~drew/admissible_5453_53236.txt 53,236]
| [http://math.mit.edu/~drew/admissible_5000_48472.txt 48,472]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_3500000_55233744.txt 55,233,744]
| [http://math.mit.edu/~drew/admissible_341640_4603276.txt 4,603,276]
| [http://math.mit.edu/~drew/admissible_181000_2326458.txt 2,326,458]
| [http://math.mit.edu/~drew/admissible_34429_388076.txt 388,076]
| [http://math.mit.edu/~drew/admissible_26024_286308.txt 286,308]
| [http://math.mit.edu/~drew/admissible_23283_253968.txt 253,968]
| [http://math.mit.edu/~drew/admissible_22949_249992.txt 249,992]
| [http://math.mit.edu/~drew/admissible_10719_108694.txt 108,694]
| [http://math.mit.edu/~drew/admissible_7140_69564.txt 69,564]
| [http://math.mit.edu/~drew/admissible_6329_60942.txt 60,942]
| [http://math.mit.edu/~drew/admissible_5453_51688.txt 51,688]
| [http://math.mit.edu/~drew/admissible_5000_46968.txt 46,968]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_3500000_55233504.txt 55,233,504]
| [http://math.mit.edu/~drew/admissible_341640_4597926.txt 4,597,926]
| [http://math.mit.edu/~drew/admissible_181000_2323344.txt 2,323,344]
| [http://math.mit.edu/~drew/admissible_34429_386344.txt 386,344]
| [http://math.mit.edu/~drew/admissible_26024_285102.txt 285,102]
| [http://math.mit.edu/~drew/admissible_23283_252720.txt 252,720]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_22949_248816_262.txt 248,816]
| [http://math.mit.edu/~drew/admissible_10719_108440.txt 108,440]
| [http://math.mit.edu/~drew/admissible_7140_69280.txt 69,280]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726]
| [http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526]
| [http://math.mit.edu/~drew/admissible_5000_46806.txt 46,806]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 56,238,957
| 4,694,650
| 2,372,232
| 394,096
| 290,604
| 257,405
| 253,381
| 110,119
| 70,496
| 61,727
| 52,371
| 47,586
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
|
|
|
|
|
|
| 197,874
| 87,690
| 56,726
| 49,794
| 42,494
| 38,710
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
|
|
|
| 301,864
| 224,100
| 198,998
| 195,962
| 86,940
| 56,238
| 49,356
| 42,114
| 38,342
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23930 29,508,018]
| 3,298,126
| 1,703,774
| 297,726
| 221,266
| 196,562
| 193,578
| 85,954
| 55,614
| 48,858
| 41,648
| 37,920
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
|
| 2,365,090
| 1,252,938
| 238,264
| 180,094
| 161,092
| 158,802
| 74,160
| 49,320
| 43,688
| 37,630
| 34,590
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
|
| 2,364,700
| 1,252,726
| 238,222
| 180,064
| 161,062
| 158,776
| 74,150
| 49,312
| 43,684
| 37,630
| 34,590
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,503,908]
| 2,751,677
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,395,694]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,872]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,420]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,691]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,298]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,314]
| 42,551
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,274]
| 31,644
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,781]
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 32,469,985]
| 2,748,330
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23913 1,393,869]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 234,529]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 173,140]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 153,447]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 151,056]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 66,211]
| 42,471
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 37,207]
| 31,584
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 28,737]
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,765,216]
| 2,677,851
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,357,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23641 227,078]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,860]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,719]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,393]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,917]
| 40,946
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,903]
| 30,478
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,708]
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 31,756,667]
| 2,676,967
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,356,644]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23634 226,987]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 167,793]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 148,656]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 146,338]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23896 63,886]
| 40,929
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 35,887]
| 30,463
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 27,696]
|-
|Brun-Titchmarsh
| 30,137,225
| 2,517,690
| 1,272,083
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23611 211,046]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 155,555]
| 137,756
| 135,599
| 58,863
| 37,610
| 32,916
| 27,910
| 25,351
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 28,080,007]
| 2,342,969
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23926 1,184,954]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 197,096]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23777 145,711]
| 128,971
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 126,931]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 55,178]
| 35,235
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23950 30,982]
| 26,388
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 24,037]
|}

{| border=1
|-
!<math>k_0</math> !! 4,000 !! 3,405 !! 3,000 !! 2,000 !! 1,783 !! 1,000 !! 672 !! 632 !! 342
|-
! Upper bounds
|-
| First <math>k_0</math> primes past <math>k_0</math>
| 39,660
| 33,222
| 28,972
| 18,386
| 16,174
| 8,424
| 5,406
| 5,028
| 2,472
|-
|Zhang sieve
| [http://math.mit.edu/~drew/admissible_4000_38596.txt 38,596]
| [http://math.mit.edu/~drew/admissible_3405_32296.txt 32,296]
| [http://math.mit.edu/~drew/admissible_3000_28008.txt 28,008]
| [http://math.mit.edu/~drew/admissible_2000_17766.txt 17,766]
| [http://math.mit.edu/~drew/admissible_1783_15620.txt 15,620]
| [http://math.mit.edu/~drew/admissible_1000_8218.txt 8,218]
| [http://math.mit.edu/~drew/admissible_672_5216.txt 5,216]
| [http://math.mit.edu/~drew/admissible_632_4860.txt 4,860]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
|Hensley-Richards sieve
| [http://math.mit.edu/~drew/admissible_4000_38498.txt 38,498]
| [http://math.mit.edu/~drew/admissible_3405_31820.txt 31,820]
| [http://math.mit.edu/~drew/admissible_3000_27806.txt 27,806]
| [http://math.mit.edu/~drew/admissible_2000_17726.txt 17,726]
| [http://math.mit.edu/~drew/admissible_1783_15756.txt 15,756]
| [http://math.mit.edu/~drew/admissible_1000_8258.txt 8,258]
| [http://math.mit.edu/~drew/admissible_672_5314.txt 5,314]
| [http://math.mit.edu/~drew/admissible_632_4918.txt 4,918]
| [http://math.mit.edu/~drew/admissible_342_2446.txt 2,446]
|-
|Asymmetric Hensley-Richards
| [http://math.mit.edu/~drew/admissible_4000_37932.txt 37,932]
| [http://math.mit.edu/~drew/admissible_3405_31762.txt 31,762]
| [http://math.mit.edu/~drew/admissible_3000_27638.txt 27,638]
| [http://math.mit.edu/~drew/admissible_2000_17676.txt 17,676]
| [http://math.mit.edu/~drew/admissible_1783_15470.txt 15,470]
| [http://math.mit.edu/~drew/admissible_1000_8168.txt 8,168]
| [http://math.mit.edu/~drew/admissible_672_5220.txt 5,220]
| [http://math.mit.edu/~drew/admissible_632_4876.txt 4,876]
| [http://math.mit.edu/~drew/admissible_342_2424.txt 2,424]
|-
|Shifted Schinzel sieve
| [http://math.mit.edu/~drew/admissible_4000_38006.txt 38,006]
| [http://math.mit.edu/~drew/admissible_3405_31910.txt 31,910]
| [http://math.mit.edu/~drew/admissible_3000_27600.txt 27,600]
| [http://math.mit.edu/~drew/admissible_2000_17554.txt 17,554]
| [http://math.mit.edu/~drew/admissible_1783_15484.txt 15,484]
| [http://math.mit.edu/~drew/admissible_1000_8072.txt 8,072]
| [http://math.mit.edu/~drew/admissible_672_5196.txt 5,196]
| [http://math.mit.edu/~drew/admissible_632_4868.txt 4,868]
| [http://math.mit.edu/~drew/admissible_342_2416.txt 2,416]
|-
| Greedy-greedy sieve
| [http://math.mit.edu/~drew/admissible_4000_36756.txt 36,756]
| [http://math.mit.edu/~drew/admissible_3405_30750.txt 30,750]
| [http://math.mit.edu/~drew/admissible_3000_26754.txt 26,754]
| [http://math.mit.edu/~drew/admissible_2000_17054.txt 17,054]
| [http://math.mit.edu/~drew/admissible_1783_15036.txt 15,036]
| [http://math.mit.edu/~drew/admissible_1000_7854.txt 7,854]
| [http://math.mit.edu/~drew/admissible_672_5030.txt 5,030]
| [http://math.mit.edu/~drew/admissible_632_4710.txt 4,710]
| [http://math.mit.edu/~drew/admissible_342_2350.txt 2,350]
|-
| Engelsma data
| [http://www.opertech.com/primes/webdata/k4000-4507/k4000-4099/k4000_36622.txt 36,622]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3400-3499/k3405_30606.txt 30,606]
| [http://www.opertech.com/primes/webdata/k3000-3999/k3000-3099/k3000_26622.txt 26,622]
| [http://www.opertech.com/primes/webdata/k2000-2999/k2000-2099/k2000_16978.txt 16,978]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1783_14958.txt 14,958]
| [http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1000_7802.txt 7,802]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4998.txt 4,998]
| [http://www.opertech.com/primes/webdata/k2-999/k600-699/k672_4680.txt 4,680]
| [http://www.opertech.com/primes/webdata/k2-999/k300-399/k342_2328.txt 2,328]
|-
|Best known tuple
| [http://math.mit.edu/~drew/admissible_4000_36610.txt 36,610]
| [http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_3000_26606_-29486.txt 26,606]
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_2000_16978_1108.txt 16,978]
| [http://math.mit.edu/~drew/admissible_1783_14950.txt 14,950]
| [http://math.mit.edu/~drew/admissible_1000_7802.txt 7,802]
| [http://math.mit.edu/~drew/admissible_672_4998.txt 4,998]
| [http://math.mit.edu/~drew/admissible_632_4680.txt '''4,680''']
| [http://math.mit.edu/~drew/admissible_342_2328.txt 2,328]
|-
! Predictions
|-
| <math>k_0 \log k_0 + k_0</math>
| 37,176
| 31,098
| 27,019
| 17,202
| 15,131
| 7,907
| 5,046
| 4,708
| 2,338
|-
! Lower bounds
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 23)
| 30,560
| 25,734
| 22,432
| 14,410
| 12,678
| 6,696
| 4,374
| 4,104
| 2,110
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 19)
| 30,366
| 25,566
| 22,284
| 14,332
| 12,614
| 6,672
| 4,344
| 4,080
| 2,096
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 17)
| 30,132
| 25,328
| 22,086
| 14,176
| 12,522
| 6,660
| 4,310
| 4,020
| 2,072
|-
|Inclusion-exclusion (<math>p_\text{exh}</math>= 13)
| 29,824
| 25,058
| 21,838
| 14,046
| 12,408
| 6,594
| 4,278
| 3,976
| 2,046
|-
|Partitioning (<math>p_\text{exh}</math>= 11)
| 27,570
| 23,532
| 20,712
| 13,730
| 12,248
| 6,812
| 4,574
| 4,278
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 7)
| 27,556
| 23,524
| 20,704
| 13,724
| 12,244
| 6,810
| 4,574
| 4,276
| 2,328
|-
|Partitioning (<math>p_\text{exh}</math>= 5)
| 27,556
| 23,524
| 20,704
| 13,722
| 12,244
| 6,808
| 4,574
| 4,276
| 2,328
|-
|MV with <math>c=1</math> (conjectural)
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,564]
| 18,898
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,456]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,500]
| 9,253
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,858]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,124]
| 2,919
| 1,454
|-
|MV with <math>c=3.2/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 22,523]
| 18,866
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 16,428]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,480]
| 9,236
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,847]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 3,118]
| 2,913
| 1,450
|-
|MV with <math>c=\sqrt{22}/\pi</math>
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,701]
| 18,153
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,758]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,061]
| 8,850
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,648]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,979]
| 2,778
| 1,361
|-
|Second Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 21,690]
| 18,143
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 15,751]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 10,056]
| 8,845
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 4,645]
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23905 2,977]
| 2,776
| 1,360
|-
|Brun-Titchmarsh
| 19,785
| 16,536
| 14,358
| 9,118
| 8,013
| 4,167
| 2,648
| 2,468
| 1,214
|-
|First Montgomery-Vaughan
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 18,859]
| 15,783
| 13,696
| [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23906 8,615]
| 7,547
| 3,959
| 2,558
| 2,392
| 1,191
|}

The '''bold number''' indicates the best currently known result for a twin-prime-like theorem.

For the Zhang tuples the minimal <math>m</math> that produced an admissible <math>k_0</math>-tuple was used. In some cases one can achieve a smaller diameter using an <math>m</math> that is slightly larger than the minimal admissible value, as noted [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23917 here].

The shifted Schinzel tuples were generated with <math>y=2</math> using an optimally chosen interval contained in <math>[-k_0\log k_0, 2k_0\log k_0]</math> (the interval is not in every case guaranteed to be optimal, particularly for larger values of <math>k_0</math>, but it is believed to be so).

The greedy-greedy tuples were generated using Sutherland's original algorithm, breaking ties downward in every case (and the optimal interval in <math>[-k_0\log k_0, 2k_0\log k_0]</math> was selected on this basis).
As [http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 noted by Castryck], breaking ties upward may produce better results in some cases.

The lower bounds listed in the partitioning and inclusion-exclusion rows were computed as described by Avishay in Sections 1 resp. 2 of this [https://sites.google.com/site/avishaytal/files/Primes.pdf document] (the case <math>k_0</math>=342 corresponds to the trivial partition). The partitioning method was strengthened by using [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24517 <math>H(343) \geq 2334</math>], [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(370) \geq 2530</math>] and [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24327 <math>H(385) \geq 2656</math>] (a complete list of bounds for <math>k_0</math> up to 4,000,000 can be found [http://math.mit.edu/~drew/partition_bounds_342_plus_4000000.txt here]), and (for <math>k_0 \leq</math> 341640) by combining the partition method with sieving for primes up to <math>p_\text{exh}</math>, as described [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24524 here].
The inclusion-exclusion involved an exhaustive search (along [http://sbseminar.wordpress.com/2013/07/02/the-quest-for-narrow-admissible-tuples/#comment-24513 these lines]) up to <math>p_\text{exh}</math>, using the inclusion-exclusion set of primes greater than <math>p_\text{exh}</math> and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.