View source for Timeline of prime gap bounds

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

127,277,395,046?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland])
| Fast admissibility testing for Hensley-Richards tuples
|}


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

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