Bounded gaps between primes: Difference between revisions

Revision as of 16:18, 22 December 2013

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

Current records

This table lists the current best bounds on [math]\displaystyle{ H_m }[/math] - the least quantity for which it is the case that there are infinitely many intervals [math]\displaystyle{ n, n+1, \ldots, n+H_m }[/math] which contain [math]\displaystyle{ m+1 }[/math] consecutive primes - both on the assumption of the Elliott-Halberstam conjecture, without this assumption, and without EH or the use of Deligne's theorems. The boldface entry - the bound on [math]\displaystyle{ H_1 }[/math] without assuming Elliott-Halberstam, but assuming the use of Deligne's theorem - is the quantity that has attracted the most attention.

[math]\displaystyle{ m }[/math]	Assuming EH	Without EH	Without EH or Deligne
1	12	272	272
2	272	395,122	493,408
3		24,490,410
4	493,408
[math]\displaystyle{ m }[/math]	[math]\displaystyle{ \displaystyle O( m e^{2m} ) }[/math]	[math]\displaystyle{ \exp((4 - \frac{52}{283} + o(1)) m) }[/math]	[math]\displaystyle{ \exp((4 - \frac{4}{43} +o(1)) m) }[/math]

Timeline of bounds

See timeline of prime gap bounds.

Polymath threads

I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013. Inactive
The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013. Inactive
Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013. Inactive
Online reading seminar for Zhang’s “bounded gaps between primes”, Terence Tao, 4 June 2013. Inactive
More narrow admissible sets, Scott Morrison, 5 June 2013. Inactive
The elementary Selberg sieve and bounded prime gaps, Terence Tao, 8 June 2013. Inactive
A combinatorial subset sum problem associated with bounded prime gaps, Terence Tao, 10 June 2013. Inactive
Further analysis of the truncated GPY sieve, Terence Tao, 11 June 2013. Inactive
Estimation of the Type I and Type II sums, Terence Tao, 12 June 2013. Inactive
Estimation of the Type III sums, Terence Tao, 14 June 2013. Inactive
A truncated elementary Selberg sieve of Pintz, Terence Tao, 18 June, 2013. Inactive
Bounding short exponential sums on smooth moduli via Weyl differencing, Terence Tao, 22 June, 2013. Inactive
The distribution of primes in densely divisible moduli, Terence Tao, 23 June, 2013. Inactive
Bounded gaps between primes (Polymath8) – a progress report, Terence Tao, 30 June 2013. Inactive
The quest for narrow admissible tuples, Andrew Sutherland, 2 July 2013. Inactive
The distribution of primes in doubly densely divisible moduli, Terence Tao, 7 July 2013. Inactive.
An improved Type I estimate, Terence Tao, 27 July 2013. Inactive
Polymath8: writing the paper, Terence Tao, 17 August 2013. Inactive
Polymath8: writing the paper, II, Terence Tao, 2 September 2013. Inactive
Polymath8: writing the paper, III, Terence Tao, 22 September 2013. Inactive
Polymath8: writing the paper, IV, Terence Tao, 15 October 2013. Inactive
Polymath8: Writing the first paper, V, and a look ahead, Terence Tao, 17 November 2013. Active
Polymath8b: Bounded intervals with many primes, after Maynard, Terence Tao, 19 November 2013. Inactive
Polymath8b, II: Optimising the variational problem and the sieve Terence Tao, 22 November 2013. Inactive
Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves, Terence Tao, 8 December 2013. Inactive
Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics, Terence Tao, 20 December 2013. Active

Writeup

Files for the draft paper for this project may be found in this directory. The compiled PDF is available here.

Here are the Polymath8 grant acknowledgments.

Code and data

Hensely-Richards sequences, Scott Morrison
- A mathematica notebook for finding k_0, Scott Morrison
- python implementation, Avi Levy
k-tuple pattern data, Thomas J Engelsma
- A graph of this data
- Tuples giving this data
Sifted sequences, Vit Tucek
Other sifted sequences, Christian Elsholtz
Size of largest admissible tuples in intervals of length up to 1050, Clark and Jarvis
C implementation of the "greedy-greedy" algorithm, Andrew Sutherland
Dropbox for sequences, pedant
Spreadsheet for admissible sequences, Vit Tucek
Java code for optimising a given tuple V1.1, xfxie
Mathematica Notebook for optimising M_k, James Maynard

Tuples applet

Here is 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 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 "Bounded gaps between primes"
1. Page 5: In the first display, [math]\displaystyle{ \mathcal{E} }[/math] should be multiplied by [math]\displaystyle{ \mathcal{L}^{2k_0+2l_0} }[/math], because [math]\displaystyle{ \lambda(n)^2 }[/math] in (2.2) can be that large, cf. (2.4).
2. Page 14: In the final display, the constraint [math]\displaystyle{ (n,d_1=1 }[/math] should be [math]\displaystyle{ (n,d_1)=1 }[/math].
3. Page 35: In the display after (10.5), the subscript on [math]\displaystyle{ {\mathcal J}_i }[/math] should be deleted.
4. Page 36: In the third display, a factor of [math]\displaystyle{ \tau(q_0r)^{O(1)} }[/math] may be needed on the right-hand side (but is ultimately harmless).
5. Page 38: In the display after (10.14), [math]\displaystyle{ \xi(r,a;q_1,b_1;q_2,b_2;n,k) }[/math] should be [math]\displaystyle{ \xi(r,a;k;q_1,b_1;q_2,b_2;n) }[/math].
6. Page 42: In (12.3), [math]\displaystyle{ B }[/math] should probably be 2.
7. Page 47: In the third display after (13.13), the condition [math]\displaystyle{ l \in {\mathcal I}_i(h) }[/math] should be [math]\displaystyle{ l \in {\mathcal I}_i(sh) }[/math].
8. Page 49: In the top line, a comma in [math]\displaystyle{ (h_1,h_2;,n_1,n_2) }[/math] should be deleted.
9. Page 51: In the penultimate display, one of the two consecutive commas should be deleted.
10. Page 54: Three displays before (14.17), [math]\displaystyle{ \bar{r_2}(m_1+m_2)q }[/math] should be [math]\displaystyle{ \bar{r_2}(m_1+m_2)/q }[/math].
Errata for Motohashi-Pintz's "A smoothed GPY sieve", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper.
1. 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]\displaystyle{ (\log \frac{R}{|D|})^{2\ell+1} }[/math], (4.15) contains instead a factor of [math]\displaystyle{ (\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]\displaystyle{ \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 "A note on bounded gaps between primes", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper.
1. Page 7: In (2.39), the exponent of [math]\displaystyle{ 3a/2 }[/math] should instead be [math]\displaystyle{ -5a/2 }[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
2. 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.

MathOverflow

Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture, 20 May 2013.
A technical question related to Zhang’s result of bounded prime gaps, 25 May 2013.
How does Yitang Zhang use Cauchy’s inequality and Theorem 2 to obtain the error term coming from the [math]\displaystyle{ S_2 }[/math] sum, 31 May 2013.
Tightening Zhang’s bound, 3 June 2013.
- Metathread for this post
Does Zhang’s theorem generalize to 3 or more primes in an interval of fixed length?, 3 June 2013.

Wikipedia and other references

Recent papers and notes

On the exponent of distribution of the ternary divisor function, Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
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. arXiv
Bounded gaps between primes, Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture, Janos Pintz, 27 May 2013.
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.
The Friedlander-Iwaniec sum, É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
Zhang's Theorem on Bounded Gaps Between Primes, Dan Goldston, May? 2013.
Notes on Zhang's prime gaps paper, Terence Tao, 1 June 2013.
Bounded prime gaps in short intervals, Johan Andersson, 3 June 2013.
Bounded length intervals containing two primes and an almost-prime II, James Maynard, 5 June 2013.
A note on bounded gaps between primes, Janos Pintz, 6 June 2013.
Lower Bounds for Admissible k-tuples, Avishay Tal, 15 June 2013.
Notes in a truncated elementary Selberg sieve (Section 1, Section 2, Section 3), Janos Pintz, 18 June 2013.
Lecture notes: bounded gaps between primes, Gergely Harcos, 1 Oct 2013.
New bounds on gaps between primes, Andrew Sutherland, 17 Oct 2013.
Bounded gaps between primes, Andrew Granville, 29 Oct 2013.
Primes in intervals of bounded length, Andrew Granville, 19 Nov 2013.
Small gaps between primes, James Maynard, 19 Nov 2013.
A note on the theorem of Maynard and Tao, Tristan Freiberg, 21 Nov 2013.
Consecutive primes in tuples, William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, 27 Nov 2013.
Close encounters among the primes, John Friedlander, Henryk Iwaniec, 10 Dec 2013.

Media

First proof that infinitely many prime numbers come in pairs, Maggie McKee, Nature, 14 May 2013.
Proof that an infinite number of primes are paired, Lisa Grossman, New Scientist, 14 May 2013.
Unheralded Mathematician Bridges the Prime Gap, Erica Klarreich, Simons science news, 20 May 2013.
- The article also appeared on Wired as "Unknown Mathematician Proves Elusive Property of Prime Numbers".
The Beauty of Bounded Gaps, Jordan Ellenberg, Slate, 22 May 2013.
Game of proofs boosts prime pair result by millions, Jacob Aron, New Scientist, 4 June 2013.
L'union fait la force des mathématiciens, Philippe Pajot, Le Monde, 24 June, 2013.
Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers, Amir Aczel, Discover Magazine, 10 July, 2013.
The Twin Prime Hero, Michael Segal, Nautilus, Issue 005, 2013.
Prime Time, Casey Hamilton, Australian National University, 19 November 2013.
Together and Alone, Closing the Prime Gap, Erica Klarreich, Quanta, 19 November 2013.
- The article also appeared on Wired as "Sudden Progress on Prime Number Problem Has Mathematicians Buzzing".
- Mathematicians Team Up To Close the Prime Gap, Slashdot, 20 November 2013.
Ein großer Schritt zum Beweis der Primzahlzwillingsvermutung, Hans Engler, Spektrum, 13 December 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. MathSciNet 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. MathSciNet 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. MathSciNet 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 MathSciNet Article
[FI1981] Fouvry, E.; Iwaniec, H. On a theorem of Bombieri-Vinogradov type., Mathematika 27 (1980), no. 2, 135–152 (1981). MathSciNet Article
[FI1983] Fouvry, E.; Iwaniec, H. Primes in arithmetic progressions. Acta Arith. 42 (1983), no. 2, 197–218. MathSciNet 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. MathSciNet JSTOR Appendix
[GPY2009] Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Y. Primes in tuples. I. Ann. of Math. (2) 170 (2009), no. 2, 819–862. arXiv 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. MathSciNet 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. MathSciNet Article
[HB1978] D. R. Heath-Brown, Hybrid bounds for Dirichlet L-functions. Invent. Math. 47 (1978), no. 2, 149–170. MathSciNet Article
[HB1986] D. R. Heath-Brown, The divisor function d3(n) in arithmetic progressions. Acta Arith. 47 (1986), no. 1, 29–56. MathSciNet 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. MathSciNet Article
[HR1973b] Hensley, Douglas; Richards, Ian, Primes in intervals. Acta Arith. 25 (1973/74), 375–391. MathSciNet Article
[MP2008] Motohashi, Yoichi; Pintz, János A smoothed GPY sieve. Bull. Lond. Math. Soc. 40 (2008), no. 2, 298–310. arXiv MathSciNet Article
[MV1973] Montgomery, H. L.; Vaughan, R. C. The large sieve. Mathematika 20 (1973), 119–134. MathSciNet Article
[M1978] Hugh L. Montgomery, The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84 (1978), no. 4, 547–567. MathSciNet 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. MathSciNet Article
[S1961] Schinzel, A. Remarks on the paper "Sur certaines hypothèses concernant les nombres premiers". Acta Arith. 7 1961/1962 1–8. MathSciNet 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. MathSciNet Article arXiv

Bounded gaps between primes: Difference between revisions

Revision as of 16:18, 22 December 2013

Contents

World records

Current records

Timeline of bounds

Polymath threads

Writeup

Code and data

Tuples applet

Errata

Other relevant blog posts

MathOverflow

Wikipedia and other references

Recent papers and notes

Media

Bibliography

Navigation menu

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@@ Line 42: / Line 42: @@
 === Timeline of bounds ===
-* <math>H = H_1</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).
+See [[timeline of prime gap bounds]].
-* <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]].)  More recent improvements on <math>k_0</math> have come from solving a [[Selberg sieve variational problem]].
-* <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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-/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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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])
-,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])
-,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])
-,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])
-,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])
-,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])
-,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)
-,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])
-,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])
-,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])
-,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])
-,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])
-,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])
-,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>
-,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])
-|
-? ([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])
-|
-([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])
-([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])
-[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])
-[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])
-#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-251889 Nielsen]])
-,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])
-,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])
-#*? ([http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/#comment-252282 Nielsen])
-,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])
-,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])
-,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])
-,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])
-,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])
-,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>
-#? ([http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/#comment-257969 Nielsen])
-,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])
-,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])
-,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])
-,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])
-| [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])
-,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])
-| A numerical precision issue was discovered in the earlier m=4 calculations
-|}
-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=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>.
 == Polymath threads ==