Finding narrow admissible tuples

For any natural number [math]\displaystyle{ k_0 }[/math], an admissible [math]\displaystyle{ k_0 }[/math]-tuple is a finite set [math]\displaystyle{ {\mathcal H} }[/math] of integers of cardinality [math]\displaystyle{ k_0 }[/math] which avoids at least one residue class modulo [math]\displaystyle{ p }[/math] for each prime [math]\displaystyle{ p }[/math]. (Note that one only needs to check those primes [math]\displaystyle{ p }[/math] of size at most [math]\displaystyle{ k_0 }[/math], so this is a finitely checkable condition.) Let [math]\displaystyle{ H(k_0) }[/math] denote the minimal diameter [math]\displaystyle{ \max {\mathcal H} - \min {\mathcal H} }[/math] of an admissible [math]\displaystyle{ k_0 }[/math]-tuple. As part of the Polymath8 project, we would like to find as good an upper bound on [math]\displaystyle{ H(k_0) }[/math] as possible for given values of [math]\displaystyle{ 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]\displaystyle{ k_0 \log k_0 + k_0 }[/math] for [math]\displaystyle{ 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]\displaystyle{ p }[/math] from an interval for a lot of primes [math]\displaystyle{ p }[/math]. Examples of sieves, in roughly increasing order of efficiency, are listed below.

Zhang sieve

The Zhang sieve uses the tuple

[math]\displaystyle{ {\mathcal H} = \{p_{m+1}, \ldots, p_{m+k_0}\} }[/math]

where [math]\displaystyle{ m }[/math] is minimized subject to staying admissible. Any [math]\displaystyle{ m }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math] yields an admissible tuple; in particular, one can just take [math]\displaystyle{ {\mathcal H} }[/math] to be the first [math]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ k_0 }[/math], but this is not optimal. Applying the prime number theorem then gives the upper bound [math]\displaystyle{ 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]\displaystyle{ {\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]\displaystyle{ {\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]\displaystyle{ i }[/math] is an integer and [math]\displaystyle{ i,m }[/math] are optimised to minimize the diameter while staying admissible.

Schinzel sieve

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

Shifted Schinzel sieve

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

Greedy sieve

Within a given interval, one sieves a single residue class [math]\displaystyle{ a \bmod p }[/math] for increasing primes [math]\displaystyle{ p=2,3,5,\ldots }[/math], with [math]\displaystyle{ a }[/math] chosen to maximize the number of survivors. Ties can be broken in a number of ways: minimize [math]\displaystyle{ a\in[0,p-1] }[/math], maximize [math]\displaystyle{ a\in [0,p-1] }[/math], minimize [math]\displaystyle{ |a-\lfloor p/2\rfloor| }[/math], or randomly. If not all residue classes modulo [math]\displaystyle{ p }[/math] are occupied by survivors, then [math]\displaystyle{ a }[/math] will be chosen so that no survivors are sieved. This necessarily occurs once [math]\displaystyle{ p }[/math] exceeds the number of survivors but typically happens much sooner. One then chooses the narrowest [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ {\mathcal H} }[/math] among the survivors (if there are fewer than [math]\displaystyle{ 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]\displaystyle{ [s,\ s+x] }[/math] using [math]\displaystyle{ y = 2 }[/math] and [math]\displaystyle{ z = \sqrt{x} }[/math] and then continuing in a greedy fashion, as proposed by Sutherland. One first optimizes the shift value [math]\displaystyle{ s }[/math] over some larger interval (e.g. [math]\displaystyle{ [-k_0\log\ k_0,\ k_0\log\ k_0] }[/math]) and then continues the sieving over primes [math]\displaystyle{ p \gt 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]\displaystyle{ [0,\ p-1] }[/math]).

Seeded greedy sieve

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

Iterated merging

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

As described by Sutherland, one can then replace [math]\displaystyle{ \mathcal{H}_1 }[/math] with [math]\displaystyle{ \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]\displaystyle{ \mathcal{H}_2 }[/math], which is typically made wider than the minimal interval containing [math]\displaystyle{ \mathcal{H}_1 }[/math] by a small factor [math]\displaystyle{ \delta }[/math] on each side (Sutherland suggests [math]\displaystyle{ \delta = 0.0025 }[/math]), and the number of failed attempts allowed while attempting to impove [math]\displaystyle{ \mathcal{H}_2 }[/math].

Eventually this process will tend to converge to particular [math]\displaystyle{ \mathcal{H}_1 }[/math] that it cannot improve (or more generally, a set of similar [math]\displaystyle{ \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.

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Iterated merging can be viewed as a form of simulated annealing. The set [math]\displaystyle{ \mathcal{S} }[/math] initially contains at least two admissible [math]\displaystyle{ k_0 }[/math]-tuples (typically many more), and as the algorithm proceeds the set [math]\displaystyle{ \mathcal{S} }[/math] converges toward [math]\displaystyle{ \mathcal{H}_1 }[/math] and the number of admissible [math]\displaystyle{ k_0 }[/math]-tuples it contains declines. One can regard the cardinality of the difference between [math]\displaystyle{ \mathcal{S} }[/math] and [math]\displaystyle{ \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]\displaystyle{ \mathcal{S} }[/math]).

Local optimizations

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

We can also shift [math]\displaystyle{ \mathcal H }[/math] to the left by removing its right end point [math]\displaystyle{ h_{k_0} }[/math] and replacing it with the greatest integer [math]\displaystyle{ h_0 \lt h_1 }[/math] that yields an admissible [math]\displaystyle{ k_0 }[/math]-tuple [math]\displaystyle{ \mathcal H' }[/math], and we can similarly shift [math]\displaystyle{ \mathcal H }[/math] to the right. The diameter of [math]\displaystyle{ \mathcal H' }[/math] need not be less than [math]\displaystyle{ \mathcal H }[/math], but if it is, it provides a useful replacement. More generally, by shifting [math]\displaystyle{ \mathcal H }[/math] repeatedly we can produce a sequence of admissible [math]\displaystyle{ 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]\displaystyle{ \mathcal H' }[/math] with smaller diameter than [math]\displaystyle{ \mathcal H }[/math].

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A more sophisticated local optimization involves a process of ``adjustment" proposed by Savitt. Let [math]\displaystyle{ \mathcal H }[/math] be an admissible [math]\displaystyle{ k_0 }[/math]-tuple. For a prime [math]\displaystyle{ p }[/math] and an integer [math]\displaystyle{ a }[/math], let [math]\displaystyle{ [a;p] }[/math] denote the residue class [math]\displaystyle{ a\bmod p }[/math], i.e. the set of integers [math]\displaystyle{ \{ x : x = a \bmod p\} }[/math]. Call [math]\displaystyle{ [a;p] }[/math] occupied if it contains an element of [math]\displaystyle{ \mathcal H }[/math].

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

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

Lower bounds

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

[math]\displaystyle{ \pi(x+y) - \pi(x) \leq F(y) }[/math] (*)

is established for some function [math]\displaystyle{ F(y) }[/math] of [math]\displaystyle{ y }[/math], the method of proof also gives a bound of the form

[math]\displaystyle{ k_0 \leq F( H(k_0)+1 ). }[/math] (**)

Indeed, if one assumes the prime tuples conjecture, any admissible [math]\displaystyle{ k_0 }[/math]-tuple of diameter [math]\displaystyle{ H }[/math] can be translated into an interval of the form [math]\displaystyle{ [x+1,x+H+1] }[/math] for some [math]\displaystyle{ x }[/math]. In the opposite direction, all known bounds of the form (*) proceed by using the fact that for [math]\displaystyle{ x\gt y }[/math], the set of primes between [math]\displaystyle{ x+1 }[/math] and [math]\displaystyle{ 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]\displaystyle{ \pi(x+y) - \pi(x) \leq (1 + o(1)) \frac{2y}{\log y} }[/math]

which then gives the lower bound

[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ k_0 (\sum_{q \leq Q} \frac{\mu^2(q)}{\phi(q)}) \leq H(k_0) + Q^2 }[/math]

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

Second Montgomery-Vaughan large sieve inequality

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

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

Benchmarks

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

[math]\displaystyle{ 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]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ k_0 }[/math]	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	59,093,364	4,923,060	2,486,370	411,946	303,558	268,544	264,460	114,814	73,448	64,182	54,516	49,586
Hensley-Richards sieve	57,554,086	4,802,222	2,422,558	402,790	297,454	262,794	258,780	112,868	72,538	63,708	53,654	48,634
Asymmetric Hensley-Richards	57,480,832	4,788,240	2,418,054	401,700	296,154	262,286	258,302	112,562	72,062	62,900	53,278	48,484
Shifted Schinzel sieve	56,789,070	4,740,846	2,396,594	399,248	294,810	260,714	256,702	112,200	71,930	62,892	53,236	48,472
Greedy-Schinzel hybrid	55,233,744	4,603,276	2,326,458	388,076	286,308	253,968	249,992	108,694	69,564	60,942	51,688	46,968
Best known tuple	55,233,504	4,597,926	2,323,344	386,344	285,102	252,720	248,816	108,440	69,280	60,726	51,526	46,806
Predictions
[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ c=1 }[/math] (conjectural)	32,503,908	2,751,677	1,395,694	234,872	173,420	153,691	151,298	66,314	42,551	37,274	31,644	28,781
MV with [math]\displaystyle{ c=3.2/\pi }[/math]	32,469,985	2,748,330	1,393,869	234,529	173,140	153,447	151,056	66,211	42,471	37,207	31,584	28,737
MV with [math]\displaystyle{ c=\sqrt{22}/\pi }[/math]	31,765,216	2,677,851	1,357,096	227,078	167,860	148,719	146,393	63,917	40,946	35,903	30,478	27,708
Montgomery-Vaughan	31,756,667	2,676,967	1,356,644	226,987	167,793	148,656	146,338	63,886	40,929	35,887	30,463	27,696
Brun-Titchmarsh	30,137,225	2,517,690	1,272,083	211,046	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

[math]\displaystyle{ k_0 }[/math]	4,000	3,405	3,000	2,000	1,783	1,000	672	632	603	342
Upper bounds
First [math]\displaystyle{ k_0 }[/math] primes past [math]\displaystyle{ 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	38,596	32,296	28,008	17,766	15,620	8,218	5,216	4,860	4,634	2,416
Hensley-Richards sieve	38,498	31,820	27,806	17,726	15,756	8,258	5,314	4,918	4,688	2,446
Asymmetric Hensley-Richards	37,932	31,762	27,638	17,676	15,470	8,168	5,220	4,876	4,672	2,424
Shifted Schinzel sieve	38,006	31,910	27,600	17,554	15,484	8,072	5,196	4,868	4,610	2,416
Greedy-Schinzel hybrid	36,756	30,750	26,754	17,054	15,036	7,854	5,030	4,710	4,452	2,350
Engelsma data	36,622	30,606	26,622	16,978	14,958	7,802	4,998	4,680	4,422	2,328
Best known tuple	36,610	30,600	26,606	16,978	14,950	7,802	4,998	4,680	4,422	2,328
Predictions
[math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ c=1 }[/math] (conjectural)	22,564	18,898	16,456	10,500	9,253	4,858	3,124	2,919	2,771	1,454
MV with [math]\displaystyle{ c=3.2/\pi }[/math]	22,523	18,866	16,428	10,480	9,236	4,847	3,118	2,913	2,765	1,450
MV with [math]\displaystyle{ c=\sqrt{22}/\pi }[/math]	21,701	18,153	15,758	10,061	8,850	4,648	2,979	2,778	2,633	1,361
Montgomery-Vaughan	21,690	18,143	15,751	10,056	8,845	4,645	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]\displaystyle{ m }[/math] that produced an admissible [math]\displaystyle{ k_0 }[/math]-tuple was used. In some cases one can achieve a smaller diameter using an [math]\displaystyle{ m }[/math] that is slightly larger than the minimal admissible value, as noted here.

The shifted Schinzel tuples were generated with [math]\displaystyle{ y=2 }[/math] using an optimally chosen interval contained in [math]\displaystyle{ [-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]\displaystyle{ 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]\displaystyle{ [-k_0\log k_0, 2k_0\log k_0] }[/math] was selected on this basis). As 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 document (the case [math]\displaystyle{ k_0 }[/math]=342 corresponds to the trivial partition). The partitioning method was strengthened by using [math]\displaystyle{ H(343) \geq 2334 }[/math], [math]\displaystyle{ H(370) \geq 2530 }[/math] and [math]\displaystyle{ H(385) \geq 2656 }[/math] (a complete list of bounds for [math]\displaystyle{ k_0 }[/math] up to 4,000,000 can be found here), and (for [math]\displaystyle{ k_0 \leq }[/math] 341640) by combining the partition method with sieving for primes up to [math]\displaystyle{ p_\text{exh} }[/math], as described here. The inclusion-exclusion involved an exhaustive search (along these lines) up to [math]\displaystyle{ p_\text{exh} }[/math], using the inclusion-exclusion set of primes greater than [math]\displaystyle{ p_\text{exh} }[/math] and less than the first prime where the depth-2 inclusion-exclusion bound is no longer positive.