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This is the home page for the Polymath8 project "bounded gaps between primes".
+
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", was 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. This project concluded with a bound of H=246, as well as additional bounds on H_m (see below).
  
 
== World records ==
 
== World records ==
  
* <math>H</math> is a quantity such that there are infinitely many pairs of consecutive primes of distance at most <math>H</math> apart.  Would like to be as small as possible (this is a primary goal of the Polymath8 project). 
+
=== Current records ===
* <math>k_0</math> is a quantity such that every admissible <math>k_0</math>-tuple has infinitely many translates which each contain at least two primes.  Would like to be as small as possible.  Improvements in <math>k_0</math> lead to improvements in <math>H</math>.  (The relationship is roughly of the form <math>H \sim k_0 \log k_0</math>; see the page on [[finding narrow admissible tuples]].) 
+
* <math>\varpi</math> is a technical parameter related to a specialized form of the [http://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture Elliott-Halberstam conjecture].  Would like to be as large as possible.  Improvements in <math>\varpi</math> lead to improvements in <math>k_0</math>, as described in the page on [[Dickson-Hardy-Littlewood theorems]]. In more recent work, the single parameter <math>\varpi</math> is replaced by a pair <math>(\varpi,\delta)</math> (in previous work we had <math>\delta=\varpi</math>).  These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on [[distribution of primes in smooth moduli]]. 
+
  
In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.
+
This table lists the current best upper bounds on <math>H_m</math> - the least quantity for which it is the case that there are infinitely many intervals <math>n, n+1, \ldots, n+H_m</math> which contain <math>m+1</math> consecutive primes - both on the assumption of the Elliott-Halberstam conjecture (or more precisely, a generalization of this conjecture, formulated as Conjecture 1 in [BFI1986]), without this assumption, and without EH or the use of Deligne's theorems.  The boldface entry - the bound on <math>H_1</math> without assuming Elliott-Halberstam, but assuming the use of Deligne's theorems - is the quantity that has attracted the most attention. The conjectured value <math>H_1=2</math> for <math>H_1</math> is the twin prime conjecture.
  
 
{| border=1
 
{| border=1
 
|-
 
|-
!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments
+
!<math>m</math>!!Conjectural!!Assuming EH!!Without EH!! Without EH or Deligne
 
|-
 
|-
| 14 May
+
|1
| 1/1,168 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])  
+
| 2
| 3,500,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
+
| [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 6] (on GEH)
| 70,000,000 ([http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Zhang])
+
[http://arxiv.org/abs/1311.4600 12] [M] (on EH only)
| All subsequent work is based on Zhang's breakthrough paper.
+
| <B>[http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]</B>
 +
| [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 246]
 
|-
 
|-
| 21 May
+
|2
|
+
| 6
|
+
| [http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355656 252] (on GEH)
| 63,374,611 ([http://mathoverflow.net/questions/131185/philosophy-behind-yitang-zhangs-work-on-the-twin-primes-conjecture/131354#131354 Lewko])
+
[http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-261984 270] (on EH only)
| 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
+
| [http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 395,106]
 +
| [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-262665 474,266]
 
|-
 
|-
| 28 May
+
|3
|
+
| 8
|  
+
| [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258528 52,116]
| 59,874,594 ([http://arxiv.org/abs/1305.6369 Trudgian])
+
| [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-258810 24,462,654]
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math>
+
| [http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-324263 32,285,928]
 
|-
 
|-
| 30 May
+
|4
|
+
| 12
|
+
| [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comments 474,266]
| 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])
+
| [http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 1,404,556,152]  
58,885,998? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23441 Tao])
+
| [http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 2,031,558,336]
 
+
59,093,364 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23444 Morrison])
+
 
+
57,554,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23448 Morrison])
+
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> and then <math>(\pm 1, \pm p_{m+1}, \ldots, \pm p_{m+k_0/2-1})</math> following [HR1973], [HR1973b], [R1974] and optimises in m
+
 
|-
 
|-
| 31 May
+
|5
|
+
| 16
| 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])
+
| [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/#comment-259813 4,137,854]
2,618,607 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
+
| [http://terrytao.wordpress.com/2014/06/19/polymath8-wrapping-up/#comment-378098 78,602,310,160]
| 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])
+
| [http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 124,840,189,042]
42,543,038 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23467 Morrison])
+
 
+
42,342,946 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23468 Morrison])
+
| Optimizes Zhang's condition <math>\omega>0</math>, and then uses an [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23465 improved bound] on <math>\delta_2</math>
+
 
|-
 
|-
| 1 Jun
+
|<math>m</math>
|
+
|<math>\displaystyle (1+o(1)) m \log m</math>
|
+
|<math>\displaystyle O( m e^{2m} )</math>
| 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])
+
|<math>O( \exp( 3.815 m) ) [BI]</math>
| Tiny improvement using the parity of <math>k_0</math>
+
|<math>O( m \exp((4 - \frac{4}{43}) m) )</math>
|-
+
|}
| 2 Jun
+
|
+
| 866,605 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
+
| 13,008,612 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23479 Morrison])
+
| Uses a [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23473 further improvement] on the quantity <math>\Sigma_2</math> in Zhang's analysis (replacing the previous bounds on <math>\delta_2</math>)
+
|-
+
| 3 Jun
+
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu])
+
| 341,640 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
+
| 4,982,086 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23512 Morrison])
+
4,802,222 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23516 Morrison])
+
| Uses a [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/ different method] to establish <math>DHL[k_0,2]</math> that removes most of the inefficiency from Zhang's method.
+
|-
+
| 4 Jun
+
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu])
+
1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu])
+
|
+
| 4,801,744 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23534 Sutherland])
+
4,788,240 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23543 Sutherland])
+
| Uses asymmetric version of the Hensley-Richards tuples
+
|-
+
| 5 Jun
+
|
+
| 34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232721 Paldi]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232732 v08ltu])
+
34,429? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232840 Tao]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232843 v08ltu]/[http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-232877 Harcos])
+
| 4,725,021 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Elsholtz])
+
4,717,560 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23562 Sutherland])
+
  
397,110? ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23563 Sutherland])
+
Unless listed below, all the above bounds were produced by the Polymath8 project.
  
4,656,298 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])
+
* [BI]: R. C. Baker, A. J. Irving, [http://arxiv.org/abs/1505.01815 Bounded intervals containing many primes]
 
+
* [M]: J. Maynard, [http://annals.math.princeton.edu/articles/8772 Small gaps between primes]
389,922 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23566 Sutherland])
+
 
+
388,310 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23571 Sutherland])
+
 
+
388,284 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23570 Castryck])
+
 
+
388,248 ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23573 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable.txt 388,188] ([http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23576 Sutherland])
+
 
+
387,982 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23588 Castryck])
+
 
+
387,974 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23591 Castryck])
+
 
+
| <math>k_0</math> bound uses the optimal Bessel function cutoff.  Originally only provisional due to neglect of the kappa error, but then it was confirmed that the kappa error was within the allowed tolerance.
+
<math>H</math> bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve
+
 
+
|-
+
| 6 Jun
+
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
+
<strike>1/552</strike> ([http://arxiv.org/abs/1306.1497 Pintz], [http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233149 Tao])
+
| <strike>60,000*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
+
 
+
<strike>52,295*</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233150 Peake])
+
 
+
<strike>11,123</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233151 Tao])
+
| 387,960 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23598 Angelveit])
+
[http://math.mit.edu/~drew/admissable_34429_387910.txt 387,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23599 Sutherland])
+
 
+
387,904 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23602 Angeltveit])
+
 
+
[http://math.mit.edu/~drew/admissable_34429_387814.txt 387,814] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23605 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable_34429_387766.txt 387,766] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23608 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable_34429_387754.txt 387,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23613 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable_34429_387620.txt 387,620] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23652 Sutherland])
+
 
+
<strike>768,534*</strike> ([http://arxiv.org/abs/1306.1497 Pintz])
+
 
+
| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve.
+
|-
+
| 7 Jun
+
| <strike>(1/538, 1/660)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
+
<strike>(1/538, 31/20444)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])
+
 
+
<strike>(1/942, 19/27004)</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])
+
 
+
<math>828 \varpi + 172\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu]/[http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-233400 Green])
+
| <strike>11,018</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233167 Tao])
+
<strike>10,721</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233178 v08ltu])
+
 
+
<strike>10,719</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233182 v08ltu])
+
 
+
<strike>25,111</strike> ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233321 v08ltu])
+
 
+
26,024? ([http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-theory-and-the-work-of-goldston-pintz-yildirim-motohashi-pintz-and-zhang/#comment-233364 vo8ltu])
+
| <strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt 113,520]?</strike> ([http://maths-people.anu.edu.au/~angeltveit/admissible_11123_113520.txt Angeltveit])
+
<strike>[http://maths-people.anu.edu.au/~angeltveit/admissible_10721_109314.txt 109,314]?</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23663 Angeltveit/Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_60000_707328.txt 707,328*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10721_108990.txt 108,990]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23666 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_11123_113462.txt 113,462*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_11018_112302.txt 112,302*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23667 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_11018_112272.txt 112,272*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23669 Sutherland])
+
 
+
<strike>116,386*</strike> ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20116 Sun])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10719_108978.txt 108,978]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23675 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10719_108634.txt 108,634]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23677 Sutherland])
+
 
+
<strike>[https://perswww.kuleuven.be/~u0040935/108632.txt 108,632]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23680 Castryck])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10719_108600.txt 108,600]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23682 Sutherland])
+
 
+
<strike>[https://perswww.kuleuven.be/~u0040935/108570.txt 108,570]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23683 Castryck])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10719_108556.txt 108,556]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23684 Sutherland])
+
 
+
<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissable_10719_108550.txt 108,550]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23688 xfxie])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_25111_275424.txt 275,424]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23694 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_10719_108540.txt 108,540]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23695 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_25111_275418.txt 275,418]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23697 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissable_25111_275404.txt 275,404]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23699 Sutherland])
+
 
+
<strike>[https://perswww.kuleuven.be/~u0040935/25111_275292.txt 275,292]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23701 Castryck-Sutherland])
+
 
+
<strike>275,262</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23703 Castryck]-[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23702 pedant]-Sutherland)
+
 
+
<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275388.txt 275,388*]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23704 xfxie]-Sutherland)
+
 
+
<strike>[https://perswww.kuleuven.be/~u0040935/25111_275126.txt 275,126]</strike> ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23706 Castryck]-pedant-Sutherland)
+
 
+
<strike>274,970</strike> ([https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Castryck-pedant-Sutherland])
+
 
+
<strike>[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt 275,208]</strike>* ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_25111_275208.txt xfxie])
+
 
+
387,534 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23716 pedant-Sutherland])
+
| Many of the results ended up being retracted due to a number of issues found in the most recent preprint of Pintz.
+
|-
+
| Jun 8
+
|
+
|
+
| [http://math.mit.edu/~drew/admissable_26024_286224.txt 286,224] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23720 Sutherland])
+
[http://math.mit.edu/~drew/admissable_26024_285810.txt 285,810] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23722 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_286216.txt 286,216] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23723 xfxie-Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable_34429_386750.txt 386,750]* ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23728 Sutherland])
+
 
+
285,752 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23725 pedant-Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissable_26024_285456.txt 285,456] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23748 Sutherland])
+
| values of <math>\varpi,\delta,k_0</math> now confirmed; most tuples available [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x on dropbox].  New bounds on <math>H</math> obtained via iterated merging using a randomized greedy sieve.
+
|-
+
| Jun 9
+
|
+
| 181,000*? ([http://arxiv.org/abs/1306.1497 Pintz])
+
| 2,530,338*? ([http://arxiv.org/abs/1306.1497 Pintz])
+
[http://math.mit.edu/~drew/admissible_26024_285278.txt 285,278] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23765 Sutherland]/[http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23763 xfxie])
+
 
+
[http://math.mit.edu/~drew/admissible_26024_285272.txt 285,272] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23779 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_26024_285248.txt 285,248] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23787 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_26024_285246.txt 285,246] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23790 xfxie-Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_26024_285232.txt 285,232] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23791 Sutherland])
+
| New bounds on <math>H</math> obtained by interleaving iterated merging with local optimizations.
+
|-
+
| Jun 10
+
|
+
| 23,283? ([http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233831 Harcos]/[http://terrytao.wordpress.com/2013/06/08/the-elementary-selberg-sieve-and-bounded-prime-gaps/#comment-233850 v08ltu])
+
| [http://math.mit.edu/~drew/admissible_26024_285210.txt 285,210] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23795 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_23283_253118.txt 253,118] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23812 xfxie])
+
 
+
[http://math.mit.edu/~drew/admissible_34429_386532.txt 386,532*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23813 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_23283_253048.txt 253,048] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23815 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_23283_252990.txt 252,990] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23817 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_23283_252976.txt 252,976] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23823 Sutherland])
+
| More efficient control of the <math>\kappa</math> error using the fact that numbers with no small prime factor are usually coprime
+
|-
+
| Jun 11
+
|
+
|
+
| [http://math.mit.edu/~drew/admissible_23283_252804.txt 252,804] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23840 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_181000_2345896.txt 2,345,896*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23846 Sutherland])
+
| More refined local "adjustment" optimizations, as detailed [http://michaelnielsen.org/polymath1/index.php?title=Finding_narrow_admissible_tuples#Local_optimizations here].
+
An issue with the <math>k_0</math> computation has been discovered, but is in the process of being repaired.
+
|-
+
| Jun 12
+
|
+
| 22,951 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/ Tao]/[http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234113 v08ltu])
+
22,949 ([http://terrytao.wordpress.com/2013/06/11/further-analysis-of-the-truncated-gpy-sieve/#comment-234157 Harcos])
+
| 249,180 ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23871 Castryck])
+
 
+
[http://math.mit.edu/~drew/admissible_22949_249046.txt 249,046] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23872 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_22949_249034.txt 249,034] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23874 Sutherland])
+
| Improved bound on <math>k_0</math> avoids the technical issue in previous computations.
+
|-
+
| Jun 13
+
|
+
|
+
|
+
 
+
[http://math.mit.edu/~drew/admissible_22949_248970.txt 248,970] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23893 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_22949_248910.txt 248,910] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23903 Sutherland])
+
|
+
|-
+
| Jun 14
+
|
+
|
+
|[http://math.mit.edu/~drew/admissible_22949_248898.txt 248,898] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23909 Sutherland])
+
|
+
|-
+
| Jun 15
+
| <math>348\varpi+68\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234670 Tao])
+
| 6,330? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234677 v08ltu])
+
6,329? ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234687 Harcos])
+
 
+
6,329 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234693 v08ltu])
+
| [http://math.mit.edu/~drew/admissible_6330_60830.txt 60,830?] ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234686 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_6329_60812.txt 60,812?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23940 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60764_-67290.txt 60,764] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23944 xfxie])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt 60,772*] ([http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60772_2836.txt xfxie])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60760_-67438.txt 60,760] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23949 xfxie])
+
| Taking more advantage of the <math>\alpha</math> convolution in the Type III sums
+
|-
+
| Jun 16
+
| <math>348\varpi+68\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234742 v08ltu])
+
 
+
<strike>155\varpi+31\delta < 1 and 220\varpi + 60\delta < 1 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234779 Tao])</strike>
+
| <strike>3,405 ([http://terrytao.wordpress.com/2013/06/14/estimation-of-the-type-iii-sums/#comment-234805 v08ltu])</strike>
+
| [http://math.mit.edu/~drew/admissible_6330_60760.txt 60,760*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_6329_60756.txt 60,756] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23951 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6330_60754_2854.txt 60,754] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23954 xfxie])
+
 
+
[http://math.mit.edu/~drew/admissible_6329_60744.txt 60,744] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23952 Sutherland])
+
 
+
<strike>[http://math.mit.edu/~drew/admissible_3405_30610.txt 30,610*] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23969 Sutherland])</strike>
+
 
+
<strike>30,606 ([http://www.opertech.com/primes/summary.txt Engelsma])</strike>
+
 
+
<strike>[http://math.mit.edu/~drew/admissible_3405_30600.txt 30,600] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23970 Sutherland])</strike>
+
| Attempting to make the Weyl differencing more efficient; unfortunately, it did not work
+
|-
+
| Jun 18
+
|
+
| 5,937? (Pintz/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz Tao]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235124 v08ltu])
+
 
+
5,672? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235135 v08ltu])
+
 
+
5,459? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235145 v08ltu])
+
 
+
5,454? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])
+
 
+
5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235150 v08ltu])
+
 
+
| [http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60740_-63166.txt 60,740] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23992 xfxie])
+
 
+
[http://math.mit.edu/~drew/admissible_6329_60732 60,732] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-23999 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_6329_60726_14.txt 60,726] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24002 xfxie]-Sutherland)
+
 
+
58,866? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])
+
 
+
[http://math.mit.edu/~drew/admissible_5937_56660.txt 56,660?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24019 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_5937_56640.txt 56,640?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24020 Sutherland])
+
 
+
53,898? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20771 Sun])
+
 
+
53,842? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20773 Sun])
+
| A new truncated sieve of Pintz virtually eliminates the influence of <math>\delta</math>
+
|-
+
| Jun 19
+
|
+
| 5,455? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235147 v08ltu])
+
 
+
5,453? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235315 v08ltu])
+
 
+
5,452? ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235316 v08ltu])
+
| [http://math.nju.edu.cn/~zwsun/admissible_5453_53774.txt 53,774?] ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20779 Sun])
+
 
+
[http://math.mit.edu/~drew/admissible_5453_51544.txt 51,544?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24022 Sutherland])
+
 
+
[http://www.cs.cmu.edu/~xfxie/project/admissible/admissible_5455_51540_4678.txt 51,540?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24050 xfxie]/[http://math.mit.edu/~drew/admissible_5455_51540.txt Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_5453_51532.txt 51,532?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24023 Sutherland])
+
 
+
[http://math.mit.edu/~drew/admissible_5453_51526.txt 51,526?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24024 Sutherland])
+
 
+
53,672*? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20837 Sun])
+
 
+
[http://math.mit.edu/~drew/admissible_5452_51520.txt 51,520?] ([http://sbseminar.wordpress.com/2013/06/05/more-narrow-admissible-sets/#comment-24060 Sutherland]/[http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-20845 Hou-Sun])
+
| Some typos in <math>\kappa_3</math> estimation had placed the 5,454 and 5,453 values of <math>k_0</math> into doubt; however other refinements have counteracted this
+
|-
+
| Jun 20
+
| <math>178\varpi + 52\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235463 Tao])
+
 
+
<math>148\varpi + 33\delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235467 Tao])
+
|
+
|
+
| Replaced "completion of sums + Weil bounds" in estimation of incomplete Kloosterman-type sums by "Fourier transform + Weyl differencing + Weil bounds", taking advantage of factorability of moduli
+
|-
+
| Jun 21
+
| <math>148\varpi + 33\delta < 1</math> ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235544 v08ltu])
+
| 1,470 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235545 v08ltu])
+
 
+
1,467 ([http://terrytao.wordpress.com/2013/06/12/estimation-of-the-type-i-and-type-ii-sums/#comment-235559 v08ltu])
+
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1470_12042.txt 12,042] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
+
 
+
[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1467_12012.txt 12,012] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
+
| Systematic tables of tuples of small length have been set up [http://www.opertech.com/primes/webdata/ here] and [http://math.mit.edu/~drew/records9.txt here] (update: As of June 27 these tables have been merged and uploaded to an [http://math.mit.edu/~primegaps/ online database] of current bounds on <math>H(k)</math> for <math>k</math> up to 5000).
+
|-
+
| Jun 22
+
|
+
| <strike>1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235740 Harcos]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235759 v08ltu])</strike>
+
| <strike>[http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])</strike>
+
| Slight improvement in the <math>\tilde \theta</math> parameter in the Pintz sieve; unfortunately, it does not seem to currently give an actual improvement to the optimal value of <math>k_0</math>
+
|-
+
| Jun 23
+
|
+
| 1,466 ([http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235891 Paldi]/[http://terrytao.wordpress.com/2013/06/18/a-truncated-elementary-selberg-sieve-of-pintz/#comment-235905 Harcos])
+
| [http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/k1466_12006.txt 12,006] ([http://www.opertech.com/primes/webdata/k1000-1999/k1400-1499/ Engelsma])
+
| An improved monotonicity formula for <math>G_{k_0-1,\tilde \theta}</math> reduces <math>\kappa_3</math> somewhat
+
|-
+
| Jun 24
+
| <math>(134 + \tfrac{2}{3}) \varpi + 28\delta \le 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
+
 
+
<math>140\varpi + 32 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236025 Tao])
+
 
+
<strike>1/88?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
+
 
+
<strike>1/74?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236039 Tao])</strike>
+
| 1,268? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-235956 v08ltu])
+
| [http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/k1268_10206.txt 10,206?] ([http://www.opertech.com/primes/webdata/k1000-1999/k1200-1299/ Engelsma])
+
| A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates
+
|-
+
| Jun 25
+
| <math>116\varpi+30\delta<1</math>? ([http://blogs.ethz.ch/kowalski/2013/06/25/a-ternary-divisor-variation Fouvry-Kowalski-Michel-Nelson]/[http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236237 Tao])
+
| 1,346? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236123 Hannes])
+
 
+
<strike>502?? ([http://terrytao.wordpress.com/2013/06/22/bounding-short-exponential-sums-on-smooth-moduli-via-weyl-differencing/#comment-236162 Trevino])</strike>
+
 
+
1,007? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236242 Hannes])
+
| [http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/k1346_10876.txt 10,876]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1300-1399/ Engelsma])
+
 
+
<strike>[http://www.opertech.com/primes/webdata/k2-999/k500-599/k502_3612.txt 3,612]?? ([http://www.opertech.com/primes/webdata/k2-999/k500-599/ Engelsma])</strike>
+
 
+
[http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/k1007_7860.txt 7,860]? ([http://www.opertech.com/primes/webdata/k1000-1999/k1000-1099/ Engelsma])
+
 
+
| Optimistic projections arise from combining the Graham-Ringrose numerology with the announced Fouvry-Kowalski-Michel-Nelson results on d_3 distribution
+
|-
+
| Jun 26
+
| <math>116\varpi + 25.5 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236346 Nielsen])
+
 
+
<math>(112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236387 Tao])
+
| 962? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236406 Hannes])
+
| [http://www.opertech.com/primes/webdata/k2-999/k900-999/k962_7470.txt 7,470]? ([http://www.opertech.com/primes/webdata/k2-999/k900-999 Engelsma])
+
| Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see [[Distribution of primes in smooth moduli|this page]], in particular optimising van der Corput in the Type I sums.  Integrated tuples page [http://math.mit.edu/~primegaps/ now online].
+
|-
+
| Jun 27
+
| <math>108\varpi + 30 \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236502 Tao])
+
| 902? ([http://terrytao.wordpress.com/2013/06/23/the-distribution-of-primes-in-densely-divisible-moduli/#comment-236507 Hannes])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_902_6966.txt 6,966]? ([http://math.mit.edu/~primegaps/ Engelsma])
+
| Improved the Type III estimates by averaging in <math>\alpha</math>; also some slight improvements to the Type II sums.  [http://math.mit.edu/~primegaps/ Tuples page] is now accepting submissions.
+
|-
+
| Jul 1
+
| <math>(93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math>? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237087 Tao])
+
|
+
873? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237160 Hannes])
+
 
+
<strike>872? ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237181 xfxie])</strike>
+
|
+
[http://math.mit.edu/~primegaps/tuples/admissible_873_6712.txt 6,712?] ([http://math.mit.edu/~primegaps/ Sutherland])
+
 
+
<strike>[http://math.mit.edu/~primegaps/tuples/admissible_872_6696.txt 6,696?] ([http://math.mit.edu/~primegaps/ Engelsma])</strike>
+
| Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions
+
|-
+
| Jul 5
+
| <math> (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta < 1</math> ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237306 Tao])
+
|
+
720 ([http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237324 xfxie]/[http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/#comment-237489 Harcos])
+
|
+
[http://math.mit.edu/~primegaps/tuples/admissible_720_5414.txt 5,414] ([http://math.mit.edu/~primegaps/ Engelsma])
+
|
+
Weakened the assumption of <math>x^\delta</math>-smoothness of the original moduli to that of double <math>x^\delta</math>-dense divisibility
+
|-
+
| Jul 10
+
| 7/600? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-238186 Tao])
+
|
+
|
+
| An in principle refinement of the van der Corput estimate based on exploiting additional averaging
+
|-
+
| Jul 19
+
| <math>(85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta < 1</math>? ([https://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239189 Tao])
+
|
+
|
+
| A more detailed computation of the Jul 10 refinement
+
|-
+
| Jul 20
+
|
+
|
+
|
+
| Jul 5 computations now [http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239251 confirmed]
+
|-
+
| Jul 27
+
|
+
| 633? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239872 Tao])
+
632? ([http://terrytao.wordpress.com/2013/07/07/the-distribution-of-primes-in-doubly-densely-divisible-moduli/#comment-239910 Harcos])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_633_4686.txt 4,686]? ([http://math.mit.edu/~primegaps/ Engelsma])
+
[http://math.mit.edu/~primegaps/tuples/admissible_632_4680.txt 4,680]? ([http://math.mit.edu/~primegaps/ Engelsma])
+
|
+
|-
+
| Jul 30
+
| <math>168\varpi + 48\delta < 1</math>**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
+
| 1,788**? ([http://terrytao.wordpress.com/2013/07/27/an-improved-type-i-estimate/#comment-240270 Tao])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_1788_14994.txt 14,994]**? ([http://math.mit.edu/~primegaps/ Sutherland])
+
| Bound obtained without using Deligne's theorems.
+
|-
+
| Aug 17
+
|
+
| 1,783**? ([http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/#comment-242205 xfxie])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_1783_14950.txt 14,950]**? ([http://math.mit.edu/~primegaps/ Sutherland])
+
|
+
|-
+
| Oct 3
+
| 13/1080?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247146 Nelson/Michel]/[http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
+
| 604?? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247155 Tao])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_604_4428.txt 4,428]?? ([http://math.mit.edu/~primegaps/ Engelsma])
+
| Found an additional variable to apply van der Corput to
+
|-
+
| Oct 11
+
| <math>\frac{1080}{13}\varpi + \frac{330}{13} \delta < 1</math>? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247766 Tao])
+
| 603? ([http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/#comment-247790 xfxie])
+
| [http://math.mit.edu/~primegaps/tuples/admissible_603_4422.txt 4,422]?([http://math.mit.edu/~primegaps/ Engelsma])
+
| Worked out the dependence on <math>\delta</math> in the Oct 3 calculation
+
|}
+
  
 +
We have been working on improving a number of other quantities, including the quantity <math>H_m</math> mentioned above:
  
Legend:
+
* <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). 
# ? - unconfirmed or conditional
+
* <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]].
# ?? - theoretical limit of an analysis, rather than a claimed record
+
* <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]].
# <nowiki>*</nowiki> - is majorized by an earlier but independent result
+
# <nowiki>**</nowiki> - bound does not rely on Deligne's theorems
+
# strikethrough - values relied on a computation that has now been retracted
+
  
 +
=== Timeline of bounds ===
  
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>.
+
A table of bounds as a function of time may be found at [[timeline of prime gap bounds]]. In this table, infinitesimal losses in <math>\delta,\varpi</math> are ignored.
  
 
== Polymath threads ==
 
== Polymath threads ==
Line 545: Line 89:
 
# [http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ Polymath8: writing the paper], Terence Tao, 17 August 2013. <I>Inactive</I>
 
# [http://terrytao.wordpress.com/2013/08/17/polymath8-writing-the-paper/ Polymath8: writing the paper], Terence Tao, 17 August 2013. <I>Inactive</I>
 
# [http://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/ Polymath8: writing the paper, II], Terence Tao, 2 September 2013. <I>Inactive</I>
 
# [http://terrytao.wordpress.com/2013/09/02/polymath8-writing-the-paper-ii/ Polymath8: writing the paper, II], Terence Tao, 2 September 2013. <I>Inactive</I>
# [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/ Polymath8: writing the paper, III], Terence Tao, 22 September 2013. <B>Active</B>
+
# [http://terrytao.wordpress.com/2013/09/22/polymath8-writing-the-paper-iii/ Polymath8: writing the paper, III], Terence Tao, 22 September 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/10/15/polymath8-writing-the-paper-iv/ Polymath8: writing the paper, IV], Terence Tao, 15 October 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/11/17/polymath8-writing-the-first-paper-v-and-a-look-ahead/ Polymath8: Writing the first paper, V, and a look ahead], Terence Tao, 17 November 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/ Polymath8b: Bounded intervals with many primes, after Maynard], Terence Tao, 19 November 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/ Polymath8b, II: Optimising the variational problem and the sieve] Terence Tao, 22 November 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/12/08/polymath8b-iii-numerical-optimisation-of-the-variational-problem-and-a-search-for-new-sieves/ Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves], Terence Tao, 8 December 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2013/12/20/polymath8b-iv-enlarging-the-sieve-support-more-efficient-numerics-and-explicit-asymptotics/ Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics], Terence Tao, 20 December 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/01/08/polymath8b-v-stretching-the-sieve-support-further/ Polymath8b, V: Stretching the sieve support further], Terence Tao, 8 January 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/ Polymath8b, VI: A low-dimensional variational problem], Terence Tao, 17 January 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/ Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further], Terence Tao, 28 January 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/02/07/new-equidistribution-estimates-of-zhang-type-and-bounded-gaps-between-primes-and-a-retrospective/ “New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective], Terence Tao, 7 February 2013. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/ Polymath8b, VIII: Time to start writing up the results?], Terence Tao, 9 February 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/ Polymath8b, IX: Large quadratic programs], Terence Tao, 21 February 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/ Polymath8b, X: Writing the paper, and chasing down loose ends], Terence Tao, 14 April 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/ Polymath 8b, XI: Finishing up the paper], Terence Tao, 17 May 2014.<I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/06/19/polymath8-wrapping-up/ Polymath8: wrapping up], Terence Tao, 19 June 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/07/20/variants-of-the-selberg-sieve-and-bounded-intervals-containing-many-primes/ Variants of the Selberg sieve, and bounded intervals containing many primes], Terence Tao, 21 July 2014. <I>Inactive</I>
 +
# [http://terrytao.wordpress.com/2014/09/30/the-bounded-gaps-between-primes-polymath-project-a-retrospective/ The "bounded gaps between primes" Polymath project - a retrospective], Terence Tao, 30 September 2014.  <B>Active</B>
  
 
== Writeup ==
 
== Writeup ==
  
Files for the draft paper for this project may be found in [https://www.dropbox.com/sh/j2r8yia6lkzk2gv/_5Sn7mNN3T this directory]. The compiled PDF is available [https://www.dropbox.com/s/16bei7l944twojr/newgap.pdf here].
+
* Files for the submitted paper for the Polymath8a project may be found in [https://www.dropbox.com/sh/0xb4xrsx4qmua7u/AABLbLyNrYktSuGsKsXjfu37a/Revised%20version this directory].
 +
** The compiled PDF for this paper is available [https://www.dropbox.com/sh/0xb4xrsx4qmua7u/AAAFh3ElzOp6jrt0MtLyQ01ca/Revised%20version/newgap.pdf here].
 +
** The paper is now on the arXiv as "[http://arxiv.org/abs/1402.0811 New equidistribution estimates of Zhang type]".
 +
** An older unabridged version of the paper may be found [http://arxiv.org/abs/1402.0811v2 here].
 +
** The initial referee report is [https://www.dropbox.com/sh/0xb4xrsx4qmua7u/AAANw1yXYBckm0Ao9aQEe-lKa/report1C.pdf here]. 
 +
** The paper has appeared at Algebra & Number Theory 8-9 (2014), 2067--2199.
 +
* Files for the draft paper for the Polymath8 retrospective may be found in [https://www.dropbox.com/sh/koxbhwvw1ysybk9/AAB1IAAjsb9kpyilhVRLLvH5a this directory].
 +
** The compiled PDF for this paper is available [https://www.dropbox.com/sh/koxbhwvw1ysybk9/AADTJ4w3yegvgTut_Tsv0Sana/retrospective.pdf here].
 +
** The paper is now on the arXiv as [http://arxiv.org/abs/1409.8361 "The "bounded gaps between primes" Polymath project - a retrospective]".
 +
** The paper has appeared at [https://www.ems-ph.org/journals/newsletter/pdf/2014-12-94.pdf Newsletter of the European Mathematics Society, December 2014, issue 94, 13--23].
 +
* Files for the draft paper for the Polymath8b project may be found in [https://www.dropbox.com/sh/uyph1zjpcirtp9b/AAC-b6Eo8GRpUHlWsC-UlKuxa this directory].
 +
** The compiled PDF for this paper is available [https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-submitted.pdf here].
 +
** The paper is now on the arXiv as [http://arxiv.org/abs/1407.4897 Variants of the Selberg sieve, and bounded intervals containing many primes]
 +
** The paper is published at [http://www.resmathsci.com/content/1/1/12 Research in the Mathematical Sciences 2014, 1:12].
  
 
Here are the [[Polymath8 grant acknowledgments]].
 
Here are the [[Polymath8 grant acknowledgments]].
Line 564: Line 138:
 
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
 
* [http://sbseminar.wordpress.com/2013/05/30/i-just-cant-resist-there-are-infinitely-many-pairs-of-primes-at-most-59470640-apart/#comment-23555 Other sifted sequences], Christian Elsholtz
 
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
 
* [http://www.ams.org/journals/mcom/2001-70-236/S0025-5718-01-01348-5/S0025-5718-01-01348-5.pdf Size of largest admissible tuples in intervals of length up to 1050], Clark and Jarvis
* [http://math.mit.edu/~drew/admissable_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
+
* [http://math.mit.edu/~drew/admissible_v0.1.tar C implementation of the "greedy-greedy" algorithm], Andrew Sutherland
 
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
 
* [https://www.dropbox.com/sh/jjxi0jmcskx1xcz/iBBVwZTj-x Dropbox for sequences], pedant
 
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
 
* [https://docs.google.com/spreadsheet/ccc?key=0Ao3urQ79oleSdEZhRS00X1FLQjM3UlJTZFRqd19ySGc&usp=sharing Spreadsheet for admissible sequences], Vit Tucek
 
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie
 
* [http://www.cs.cmu.edu/~xfxie/project/admissible/admissibleV1.1.zip Java code for optimising a given tuple V1.1], xfxie
 +
* [https://math.mit.edu/~primegaps/MaynardMathematicaNotebook.txt Mathematica Notebook for optimising M_k], James Maynard
 +
* Some [[notes on polytope decomposition]]
 +
* [https://math.mit.edu/~drew/ompadm_v0.5.tar Multi-threaded admissibility testing for very large tuples], Andrew Sutherland
 +
* [http://users.ugent.be/~ibogaert/KrylovMk/KrylovMk.pdf Krylov method for lower bounding M_k], Ignace Bogaert
  
 
=== Tuples applet ===
 
=== Tuples applet ===
Line 581: Line 159:
 
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).
 
The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).
  
A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and I have only tested it on the current versions of Chrome and Firefox (it should work in most browsers but I make no guarantees). Also, the MIT math web server has recently developed the habit of occasionally throwing 403 errors when you try to load a page — if this happens just hit reload.
+
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 ==
 
== Errata ==
Line 587: Line 165:
 
Page numbers refer to the file linked to for the relevant paper.
 
Page numbers refer to the file linked to for the relevant paper.
  
# Errata for Zhang's "[http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes]"
+
# Errata for Zhang's "[http://annals.math.princeton.edu/2014/179-3/p07 Bounded gaps between primes]"
 
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
 
## Page 5: In the first display, <math>\mathcal{E}</math> should be multiplied by <math>\mathcal{L}^{2k_0+2l_0}</math>, because <math>\lambda(n)^2</math> in (2.2) can be that large, cf. (2.4).
 
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
 
## Page 14: In the final display, the constraint <math>(n,d_1=1</math> should be <math>(n,d_1)=1</math>.
Line 619: Line 197:
 
* [http://blogs.ethz.ch/kowalski/2013/09/09/conductors-of-one-variable-transforms-of-trace-functions/ Conductors of one-variable transforms of trace functions], Emmanuel Kowalski, 9 September 2013.
 
* [http://blogs.ethz.ch/kowalski/2013/09/09/conductors-of-one-variable-transforms-of-trace-functions/ Conductors of one-variable transforms of trace functions], Emmanuel Kowalski, 9 September 2013.
 
* [http://gilkalai.wordpress.com/2013/09/20/polymath-8-a-success/ Polymath 8 – a Success!], Gil Kalai, 20 September 2013.
 
* [http://gilkalai.wordpress.com/2013/09/20/polymath-8-a-success/ Polymath 8 – a Success!], Gil Kalai, 20 September 2013.
 +
* [http://blogs.ethz.ch/kowalski/2013/10/24/james-maynard-auteur-du-theoreme-de-lannee/ James Maynard, auteur du théorème de l’année], Emmanuel Kowalski, 24 October 2013.
 +
* [http://blogs.ethz.ch/kowalski/2013/12/08/reflections-on-reading-the-polymath8a-paper/ Reflections on reading the Polymath8(a) paper], Emmanuel Kowalski, 8 December 2013.
  
 
== MathOverflow ==
 
== MathOverflow ==
Line 643: Line 223:
 
* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
 
* [http://arxiv.org/abs/1304.3199 On the exponent of distribution of the ternary divisor function], Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, 11 Apr 2013.
 
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
 
* [http://www.renyi.hu/~revesz/ThreeCorr0grey.pdf On the optimal weight function in the Goldston-Pintz-Yildirim method for finding small gaps between consecutive primes], Bálint Farkas, János Pintz and Szilárd Gy. Révész, To appear in: Paul Turán Memorial Volume: Number Theory, Analysis and Combinatorics, de Gruyter, Berlin, 2013. 23 pages. [http://arxiv.org/abs/1306.2133 arXiv]
* [http://annals.math.princeton.edu/wp-content/uploads/YitangZhang.pdf Bounded gaps between primes], Yitang Zhang, to appear, Annals of Mathematics. Released 21 May, 2013.
+
* [http://arxiv.org/abs/1305.0348 The existence of small prime gaps in subsets of the integers], Jacques Benatar, 2 May, 2013.
 +
* [http://annals.math.princeton.edu/2014/179-3/p07 Bounded gaps between primes], Yitang Zhang, Annals of Mathematics 179 (2014), 1121-1174. Released 21 May, 2013.
 
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
 
* [http://arxiv.org/abs/1305.6289 Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture], Janos Pintz, 27 May 2013.
 
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
 
* [http://arxiv.org/abs/1305.6369 A poor man's improvement on Zhang's result: there are infinitely many prime gaps less than 60 million], T. S. Trudgian, 28 May 2013.
 
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
 
* [http://www.math.ethz.ch/~kowalski/friedlander-iwaniec-sum.pdf The Friedlander-Iwaniec sum], É. Fouvry, E. Kowalski, Ph. Michel., May 2013.
 +
* [http://www.aimath.org/news/primegaps70m/ Zhang's Theorem on Bounded Gaps Between Primes], Dan Goldston, May? 2013.
 
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
 
* [http://terrytao.files.wordpress.com/2013/06/bounds.pdf Notes on Zhang's prime gaps paper], Terence Tao, 1 June 2013.
 
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
 
* [http://arxiv.org/abs/1306.0511 Bounded prime gaps in short intervals], Johan Andersson, 3 June 2013.
Line 654: Line 236:
 
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.
 
* Notes in a truncated elementary Selberg sieve ([http://terrytao.files.wordpress.com/2013/06/file-1.pdf Section 1], [http://terrytao.files.wordpress.com/2013/06/file-2.pdf Section 2], [http://terrytao.files.wordpress.com/2013/06/file-3.pdf Section 3]), Janos Pintz, 18 June 2013.
 
* [http://www.renyi.hu/~gharcos/gaps.pdf Lecture notes: bounded gaps between primes], Gergely Harcos, 1 Oct 2013.
 
* [http://www.renyi.hu/~gharcos/gaps.pdf Lecture notes: bounded gaps between primes], Gergely Harcos, 1 Oct 2013.
 +
* [http://math.mit.edu/~drew/PrimeGaps.pdf New bounds on gaps between primes], Andrew Sutherland, 17 Oct 2013.
 +
* [http://www.dms.umontreal.ca/~andrew/CurrentEventsArticle.pdf Bounded gaps between primes], Andrew Granville, 29 Oct 2013.
 +
* [http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf Primes in intervals of bounded length], Andrew Granville, 19 Nov 2013.  [http://arxiv.org/abs/1410.8400 Uploaded to arXiv], 30 Oct 2014.
 +
* [http://annals.math.princeton.edu/articles/8772 Small gaps between primes], James Maynard, 19 Nov 2013.  To appear, Annals Math.
 +
* [http://arxiv.org/abs/1311.5319 A note on the theorem of Maynard and Tao], Tristan Freiberg, 21 Nov 2013.
 +
* [http://arxiv.org/abs/1311.7003 Consecutive primes in tuples],  William D. Banks, Tristan Freiberg, and Caroline L. Turnage-Butterbaugh, 27 Nov 2013.
 +
* [http://arxiv.org/abs/1312.2926 Close encounters among the primes], John Friedlander, Henryk Iwaniec, 10 Dec 2013.
 +
* [http://arxiv.org/abs/1401.7555 A Friendly Intro to Sieves with a Look Towards Recent Progress on the Twin Primes Conjecture], David Lowry-Duda, 25 Jan, 2014.
 +
* [http://arxiv.org/abs/1401.6614 The twin prime conjecture], Yoichi Motohashi, 26 Jan 2014.
 +
* [http://arxiv.org/abs/1401.6677 Bounded gaps between primes in Chebotarev sets], Jesse Thorner, 26 Jan 2014.
 +
* [http://arxiv.org/abs/1402.4849 Bounded gaps between primes], Ben Green, 19 Feb 2014.
 +
* [http://arxiv.org/abs/1403.4527 Bounded gaps between primes of the special form], Hongze Li, Hao Pan, 19 Mar 2014.
 +
* [http://arxiv.org/abs/1403.5808 Bounded gaps between primes in number fields and function fields], Abel Castillo, Chris Hall, Robert J. Lemke Oliver, Paul Pollack, Lola Thompson, 23 Mar 2014.
 +
* [http://arxiv.org/abs/1404.4007 Bounded gaps between primes with a given primitive root], Paul Pollack, 15 Apr 2014.
 +
* [http://arxiv.org/abs/1404.5094 On limit points of the sequence of normalized prime gaps], William D. Banks, Tristan Freiberg, and James Maynard, 21 Apr 2014.
 +
* [http://smf4.emath.fr/Publications/Gazette/2014/140/smf_gazette_140_19-31.pdf Petits écarts entre nombres premiers et polymath : une nouvelle manière de faire de la recherche en mathématiques?], R. de la Breteche, Gazette des Mathématiciens, Soc. Math. France, Avril 2014, 19--31.
 +
* [http://arxiv.org/abs/1405.2593 Dense clusters of primes in subsets], James Maynard, 11 May 2014.
 +
* [http://arxiv.org/abs/1405.4444 Arithmetic functions at consecutive shifted primes], Paul Pollack, Lola Thompson, 17 May 2014.
 +
* [http://arxiv.org/abs/1406.2658 On the ratio of consecutive gaps between primes], Janos Pintz, 10 Jun 2014.
 +
* [http://arxiv.org/abs/1407.1747 Bounded gaps between primes in special sequences], Lynn Chua, Soohyun Park, Geoffrey D. Smith, 8 Jul 2014.
 +
* [http://arxiv.org/abs/1407.2213 On the distribution of gaps between consecutive primes], Janos Pintz, 8 Jul 2014 (first version), 24 Sep 2014 (second version).
 +
* [http://arxiv.org/abs/1408.5110 Large gaps between primes], James Maynard, 21 Aug 2014.
 +
* [http://arxiv.org/abs/1410.8198 Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-Tao], Andrew Granville, Daniel M. Kane, Dimitris Koukoulopoulos, Robert J. Lemke Oliver, 29 Oct, 2014.
 +
* [http://arxiv.org/abs/1411.2989 Gaps between Primes in Beatty Sequences], Roger Baker, Liangyi Zhao, 11 Nov 2014.
 +
* [http://arxiv.org/abs/1501.06690 On the Density of Weak Polignac Numbers], Stijn Hanson, 27 Jan 2015.
 +
* [http://arxiv.org/abs/1504.06860 On a conjecture of Erdős, Pólya and Turán on consecutive gaps between primes], Janos Pintz, 26 Apr 2015.
 +
* [http://arxiv.org/abs/1505.01815 Bounded intervals containing many primes], R. C. Baker, A. J. Irving, 7 May 2015.
 +
* [http://arxiv.org/abs/1505.03104 Goldbach versus de Polignac numbers], Jacques Benatar, 12 May 2015.
 +
* [http://www.ams.org/notices/201506/rnoti-p660.pdf Prime numbers: A much needed gap is finally found], John Friedlander, June 2015.
 +
* [http://math.mit.edu/~drew/PrimeGapsOberwolfach1.pdf Sieve theory and small gaps between primes: Introduction], [http://math.mit.edu/~drew/PrimeGapsOberwolfach2.pdf A variational problem], [http://math.mit.edu/~drew/PrimeGapsOberwolfach3.pdf Narrow admissible tuples]  Andrew V. Sutherland, July 2015.
 +
* "Small gaps between the set of primes and products of two primes", Keiju Sono, August 2015.
 +
* [http://arxiv.org/abs/1509.01564 Patterns of primes in arithmetic progressions], Janos Pintz, 4 Sep 2015.
 +
* [http://arxiv.org/abs/1510.04577 A note on the distribution of normalized prime gaps], Janos Pintz, 15 Oct 2015.
 +
* [http://arxiv.org/abs/1510.08054 Limit points and long gaps between primes], Roger Baker, Tristan Freiberg, 27 Oct 2015.
 +
* [http://www.mast.queensu.ca/~akshaa/gaussian.pdf Bounded gaps between Gaussian primes], Akshaa Vatwani, 3 Nov 2015.
 +
* [http://arxiv.org/abs/1512.01470 General divisor functions in arithmetic progressions to large moduli], Fei Wei, Boqing Xue, Yitang Zhang, Dec 2015.
 +
* [http://arxiv.org/abs/1512.03936 Large gaps between consecutive primes containing perfect k-th powers of prime numbers], Helmut Maier, Michael Rassias, 12 Dec 2015.
 +
* [http://arxiv.org/abs/1604.01761 Increasing and decreasing prime gaps], Daniel Shiu, 6 Apr 2016.
 +
* [http://arxiv.org/abs/1604.06903 Golomb's conjecture on prime gaps], Christian Elsholtz, 23 Apr 2016.
 +
* Factors of Carmichael numbers and a weak k-tuples conjecture, Thomas Wright, J. Aust. Math. Soc. 100 (2016), 421-429.
 +
* [https://arxiv.org/abs/1605.02920 Small gaps between the set of products of at most two primes], Keiju Sono, May 10 2016.
 +
* [http://arxiv.org/abs/1607.03887 Bounded Gaps Between Products of Distinct Primes], Yang Liu, Peter S. Park, Zhuo Qun Song, Jul 14 2016.
 +
* [https://arxiv.org/abs/1705.08034 Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds], Benjamin Linowitz, D. B. McReynolds, Paul Pollack, Lola Thompson, May 24, 2017.
 +
* [https://arxiv.org/abs/1707.05437 Bounded gaps between primes in short intervals], Ryan Alweiss, Sammy Luo, Jul 18 2017.
 +
* [https://arxiv.org/abs/1711.01949 Bounded gaps between product of two primes in number fields], Pranendu Darbar, Anirban Mukhopadhyay, Nov 6 2017.
 +
* [http://arxiv.org/abs/1802.10327 Short intervals containing a prescribed number of primes], Daniele Mastrostefano, Mar 1 2018.
 +
* [http://arxiv.org/abs/1806.09034 Almost primes in various settings], Paweł Lewulis, Jun 23 2018.
 +
* [https://arxiv.org/abs/1907.02246 On the largest square divisor of shifted primes], Jori Merikoski, Jul 4 2019.
  
 
== Media ==
 
== Media ==
Line 666: Line 296:
 
* [http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/ Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers], Amir Aczel, Discover Magazine, 10 July, 2013.
 
* [http://blogs.discovermagazine.com/crux/2013/07/10/primal-madness-mathematicians-hunt-for-twin-prime-numbers/ Primal Madness: Mathematicians’ Hunt for Twin Prime Numbers], Amir Aczel, Discover Magazine, 10 July, 2013.
 
* [http://nautil.us/issue/5/fame/the-twin-prime-hero The Twin Prime Hero], Michael Segal, Nautilus, Issue 005, 2013.
 
* [http://nautil.us/issue/5/fame/the-twin-prime-hero The Twin Prime Hero], Michael Segal, Nautilus, Issue 005, 2013.
 +
* [http://news.anu.edu.au/2013/11/19/prime-time/ Prime Time], Casey Hamilton, Australian National University, 19 November 2013.
 +
* [https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ Together and Alone, Closing the Prime Gap], Erica Klarreich, Quanta, 19 November 2013.
 +
** The article also appeared on Wired as "[http://www.wired.com/wiredscience/2013/11/prime/ Sudden Progress on Prime Number Problem Has Mathematicians Buzzing]".
 +
** [http://science.slashdot.org/story/13/11/20/1256229/mathematicians-team-up-to-close-the-prime-gap Mathematicians Team Up To Close the Prime Gap], Slashdot, 20 November 2013.
 +
* [http://www.spektrum.de/alias/mathematik/ein-grosser-schritt-zum-beweis-der-primzahlzwillingsvermutung/1216488 Ein großer Schritt zum Beweis der Primzahlzwillingsvermutung], Hans Engler, Spektrum, 13 December 2013.
 +
* [http://phys.org/news/2014-01-mathematical-puzzle-unraveled.html An old mathematical puzzle soon to be unraveled?], Benjamin Augereau, Phys.org, 15 January 2014.
 +
* [http://www.spektrum.de/alias/zahlentheorie/neuer-durchbruch-auf-dem-weg-zur-primzahlzwillingsvermutung/1222001 Neuer Durchbruch auf dem Weg zur Primzahlzwillingsvermutung], Christoph Poppe, Spektrum, 30 January 2014.
 +
* [http://news.cnet.com/8301-17938_105-57618696-1/yitang-zhang-a-prime-number-proof-and-a-world-of-persistence/ Yitang Zhang: A prime-number proof and a world of persistence], Leslie Katz, CNET, February 12, 2014.
 +
* [http://podacademy.org/podcasts/maths-isnt-standing-still/ Maths isn’t standing still], Adam Smith and Vicky Neale, Pod Academy, March 3, 2014.
 +
* [https://www.quantamagazine.org/20141210-prime-gap-grows-after-decades-long-lull/ Prime Gap Grows After Decades-Long Lull], Erica Klarreich, Quanta, Dec 10, 2014.
 +
* [http://www.zalafilms.com/films/countingindex.html Counting from infinity: Yitang Zhang and the twin prime conjecture] (Documentary), George Csicsery, released Jan 2015.
 +
* [http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty The Pursuit of Beauty], Alec Wilkinson, New Yorker, Feb 2 2015.
 +
** [http://video.newyorker.com/watch/annals-of-ideas-yitang-zhang-s-discovery-2015-01-28 Yitang Zhang's discovery] (Video), Alec Wilkinson, New Yorker, Jan 28, 2015.
 +
* [http://digitaleditions.walsworthprintgroup.com/publication/?i=247647&p=16 Prime Progress Invigorates Math Minds], Katherine Merow, MAA Focus, February/March 2015.
 +
* [http://www.ams.org/notices/201506/rnoti-p660.pdf Prime Numbers: A Much Needed Gap Is Finally Found], John Friedlander, Notices of the AMS, June/July 2015.
  
 
== Bibliography ==
 
== Bibliography ==

Latest revision as of 07:51, 11 July 2019

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", was 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. This project concluded with a bound of H=246, as well as additional bounds on H_m (see below).

World records

Current records

This table lists the current best upper bounds on [math]H_m[/math] - the least quantity for which it is the case that there are infinitely many intervals [math]n, n+1, \ldots, n+H_m[/math] which contain [math]m+1[/math] consecutive primes - both on the assumption of the Elliott-Halberstam conjecture (or more precisely, a generalization of this conjecture, formulated as Conjecture 1 in [BFI1986]), without this assumption, and without EH or the use of Deligne's theorems. The boldface entry - the bound on [math]H_1[/math] without assuming Elliott-Halberstam, but assuming the use of Deligne's theorems - is the quantity that has attracted the most attention. The conjectured value [math]H_1=2[/math] for [math]H_1[/math] is the twin prime conjecture.

[math]m[/math] Conjectural Assuming EH Without EH Without EH or Deligne
1 2 6 (on GEH)

12 [M] (on EH only)

246 246
2 6 252 (on GEH)

270 (on EH only)

395,106 474,266
3 8 52,116 24,462,654 32,285,928
4 12 474,266 1,404,556,152 2,031,558,336
5 16 4,137,854 78,602,310,160 124,840,189,042
[math]m[/math] [math]\displaystyle (1+o(1)) m \log m[/math] [math]\displaystyle O( m e^{2m} )[/math] [math]O( \exp( 3.815 m) ) [BI][/math] [math]O( m \exp((4 - \frac{4}{43}) m) )[/math]

Unless listed below, all the above bounds were produced by the Polymath8 project.

We have been working on improving a number of other quantities, including the quantity [math]H_m[/math] mentioned above:

  • [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).
  • [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 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.

Timeline of bounds

A table of bounds as a function of time may be found at timeline of prime gap bounds. In this table, infinitesimal losses in [math]\delta,\varpi[/math] are ignored.

Polymath threads

  1. I just can’t resist: there are infinitely many pairs of primes at most 59470640 apart, Scott Morrison, 30 May 2013. Inactive
  2. The prime tuples conjecture, sieve theory, and the work of Goldston-Pintz-Yildirim, Motohashi-Pintz, and Zhang, Terence Tao, 3 June 2013. Inactive
  3. Polymath proposal: bounded gaps between primes, Terence Tao, 4 June 2013. Inactive
  4. Online reading seminar for Zhang’s “bounded gaps between primes”, Terence Tao, 4 June 2013. Inactive
  5. More narrow admissible sets, Scott Morrison, 5 June 2013. Inactive
  6. The elementary Selberg sieve and bounded prime gaps, Terence Tao, 8 June 2013. Inactive
  7. A combinatorial subset sum problem associated with bounded prime gaps, Terence Tao, 10 June 2013. Inactive
  8. Further analysis of the truncated GPY sieve, Terence Tao, 11 June 2013. Inactive
  9. Estimation of the Type I and Type II sums, Terence Tao, 12 June 2013. Inactive
  10. Estimation of the Type III sums, Terence Tao, 14 June 2013. Inactive
  11. A truncated elementary Selberg sieve of Pintz, Terence Tao, 18 June, 2013. Inactive
  12. Bounding short exponential sums on smooth moduli via Weyl differencing, Terence Tao, 22 June, 2013. Inactive
  13. The distribution of primes in densely divisible moduli, Terence Tao, 23 June, 2013. Inactive
  14. Bounded gaps between primes (Polymath8) – a progress report, Terence Tao, 30 June 2013. Inactive
  15. The quest for narrow admissible tuples, Andrew Sutherland, 2 July 2013. Inactive
  16. The distribution of primes in doubly densely divisible moduli, Terence Tao, 7 July 2013. Inactive.
  17. An improved Type I estimate, Terence Tao, 27 July 2013. Inactive
  18. Polymath8: writing the paper, Terence Tao, 17 August 2013. Inactive
  19. Polymath8: writing the paper, II, Terence Tao, 2 September 2013. Inactive
  20. Polymath8: writing the paper, III, Terence Tao, 22 September 2013. Inactive
  21. Polymath8: writing the paper, IV, Terence Tao, 15 October 2013. Inactive
  22. Polymath8: Writing the first paper, V, and a look ahead, Terence Tao, 17 November 2013. Inactive
  23. Polymath8b: Bounded intervals with many primes, after Maynard, Terence Tao, 19 November 2013. Inactive
  24. Polymath8b, II: Optimising the variational problem and the sieve Terence Tao, 22 November 2013. Inactive
  25. Polymath8b, III: Numerical optimisation of the variational problem, and a search for new sieves, Terence Tao, 8 December 2013. Inactive
  26. Polymath8b, IV: Enlarging the sieve support, more efficient numerics, and explicit asymptotics, Terence Tao, 20 December 2013. Inactive
  27. Polymath8b, V: Stretching the sieve support further, Terence Tao, 8 January 2014. Inactive
  28. Polymath8b, VI: A low-dimensional variational problem, Terence Tao, 17 January 2014. Inactive
  29. Polymath8b, VII: Using the generalised Elliott-Halberstam hypothesis to enlarge the sieve support yet further, Terence Tao, 28 January 2014. Inactive
  30. “New equidistribution estimates of Zhang type, and bounded gaps between primes” – and a retrospective, Terence Tao, 7 February 2013. Inactive
  31. Polymath8b, VIII: Time to start writing up the results?, Terence Tao, 9 February 2014. Inactive
  32. Polymath8b, IX: Large quadratic programs, Terence Tao, 21 February 2014. Inactive
  33. Polymath8b, X: Writing the paper, and chasing down loose ends, Terence Tao, 14 April 2014. Inactive
  34. Polymath 8b, XI: Finishing up the paper, Terence Tao, 17 May 2014.Inactive
  35. Polymath8: wrapping up, Terence Tao, 19 June 2014. Inactive
  36. Variants of the Selberg sieve, and bounded intervals containing many primes, Terence Tao, 21 July 2014. Inactive
  37. The "bounded gaps between primes" Polymath project - a retrospective, Terence Tao, 30 September 2014. Active

Writeup

Here are the Polymath8 grant acknowledgments.

Code and data

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.

  1. Errata for Zhang's "Bounded gaps between primes"
    1. Page 5: In the first display, [math]\mathcal{E}[/math] should be multiplied by [math]\mathcal{L}^{2k_0+2l_0}[/math], because [math]\lambda(n)^2[/math] in (2.2) can be that large, cf. (2.4).
    2. Page 14: In the final display, the constraint [math](n,d_1=1[/math] should be [math](n,d_1)=1[/math].
    3. Page 35: In the display after (10.5), the subscript on [math]{\mathcal J}_i[/math] should be deleted.
    4. Page 36: In the third display, a factor of [math]\tau(q_0r)^{O(1)}[/math] may be needed on the right-hand side (but is ultimately harmless).
    5. Page 38: In the display after (10.14), [math]\xi(r,a;q_1,b_1;q_2,b_2;n,k)[/math] should be [math]\xi(r,a;k;q_1,b_1;q_2,b_2;n)[/math].
    6. Page 42: In (12.3), [math]B[/math] should probably be 2.
    7. Page 47: In the third display after (13.13), the condition [math]l \in {\mathcal I}_i(h)[/math] should be [math]l \in {\mathcal I}_i(sh)[/math].
    8. Page 49: In the top line, a comma in [math](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]\bar{r_2}(m_1+m_2)q[/math] should be [math]\bar{r_2}(m_1+m_2)/q[/math].
  2. 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](\log \frac{R}{|D|})^{2\ell+1}[/math], (4.15) contains instead a factor of [math](\log \frac{R/w}{|K|})^{2\ell+1}[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\exp(-k\omega/3)[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14).
  3. 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]3a/2[/math] should instead be [math]-5a/2[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper.
    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.

Other relevant blog posts

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Recent papers and notes

Media

Bibliography

Additional links for some of these references (e.g. to arXiv versions) would be greatly appreciated.

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