Timeline of prime gap bounds: Difference between revisions
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!Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments | !Date!!<math>\varpi</math> or <math>(\varpi,\delta)</math>!! <math>k_0</math> !! <math>H</math> !! Comments | ||
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| 10 | | Aug 10 2005 | ||
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| 6 [EH] | | 6 [EH] | ||
| Line 15: | Line 9: | ||
| First bounded prime gap result (conditional on Elliott-Halberstam) | | First bounded prime gap result (conditional on Elliott-Halberstam) | ||
|- | |- | ||
| 14 | | May 14 2013 | ||
| 1/1,168 ([http://annals.math.princeton.edu/ | | 1/1,168 ([http://annals.math.princeton.edu/articles/7954 Zhang]) | ||
| 3,500,000 ([http://annals.math.princeton.edu/ | | 3,500,000 ([http://annals.math.princeton.edu/articles/7954 Zhang]) | ||
| 70,000,000 ([http://annals.math.princeton.edu/ | | 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. | | All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper. | ||
|- | |- | ||
| 21 | | May 21 | ||
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| Line 27: | Line 21: | ||
| 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 | | 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 | ||
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| 28 | | May 28 | ||
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| Line 33: | Line 27: | ||
| Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math> | | Uses <math>(p_{m+1},\ldots,p_{m+k_0})</math> with <math>p_{m+1} > k_0</math> | ||
|- | |- | ||
| 30 | | May 30 | ||
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| Line 44: | Line 38: | ||
| 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 | | 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 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]) | | 2,947,442 ([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]) | ||
| Line 54: | Line 48: | ||
| 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> | | 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 1 | ||
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| Line 60: | Line 54: | ||
| Tiny improvement using the parity of <math>k_0</math> | | Tiny improvement using the parity of <math>k_0</math> | ||
|- | |- | ||
| 2 | | 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]) | | 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]) | ||
| Line 66: | Line 60: | ||
| 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>) | | 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 3 | ||
| 1/1,040? ([http://mathoverflow.net/questions/132632/tightening-zhangs-bound-closed v08ltu]) | | 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]) | | 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]) | ||
| Line 73: | Line 67: | ||
| 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. | | 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 4 | ||
| 1/224?? ([http://polymathprojects.org/2013/06/04/polymath-proposal-bounded-gaps-between-primes/#comment-19961 v08ltu]) | | 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]) | 1/240?? ([http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-bounded-gaps-between-primes/#comment-232661 v08ltu]) | ||
| Line 81: | Line 75: | ||
| Uses asymmetric version of the Hensley-Richards tuples | | Uses asymmetric version of the Hensley-Richards tuples | ||
|- | |- | ||
| 5 | | 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-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]) | ||
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| 6 | | Jun 6 | ||
| <strike>(1/488,3/9272)</strike> ([http://arxiv.org/abs/1306.1497 Pintz]) | | <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>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]) | ||
| Line 135: | Line 129: | ||
| Improved <math>H</math>-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve. | | 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 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, 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>(1/538, 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]) | ||
| Line 736: | Line 730: | ||
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]) | 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 | 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_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]) | ||
| Line 752: | Line 754: | ||
[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_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 | | A numerical precision issue was discovered in the earlier m=4 calculations | ||
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| Dec 23 | |||
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| 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]) | |||
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| Dec 24 | |||
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| [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]) | |||
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| Dec 28 | |||
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| [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]) | |||
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| Jan 2 2014 | |||
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| [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]) | |||
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| Jan 6 | |||
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| 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]) | |||
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| Jan 8 | |||
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| 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]. | |||
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| Jan 9 | |||
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| [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]) | |||
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| Jan 28 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/admissible_35146_395106.txt 395,106]? [m=2] ([http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268356 Sutherland]) | |||
| | |||
|- | |||
| Jan 29 | |||
| | |||
| 3 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen]) | |||
| 6 [GEH] ([http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 Nielsen]) | |||
| A new idea of Maynard exploits GEH to allow for cutoff functions whose support extends beyond the unit cube | |||
|- | |||
| Feb 9 | |||
| | |||
| | |||
| | |||
| Jan 29 results confirmed [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-270631 here] | |||
|- | |||
| Feb 17 | |||
| | |||
| 53?# ([http://terrytao.wordpress.com/2014/02/09/polymath8b-viii-time-to-start-writing-up-the-results/#comment-271862 Nielsen]) | |||
| 264?# ([http://math.mit.edu/~primegaps/tuples/admissible_53_264.txt Clark-Jarvis]) | |||
| Managed to get the epsilon trick to be computationally feasible for medium k | |||
|- | |||
| Feb 22 | |||
| | |||
| 51?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-272506 Nielsen]) | |||
| 252?# ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis]) | |||
| More efficient matrix computation allows for higher degrees to be used | |||
|- | |||
| Mar 4 | |||
| | |||
| | |||
| | |||
| Jan 6 computations [http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-273967 confirmed] | |||
|- | |||
| Apr 14 | |||
| | |||
| 50?# ([http://terrytao.wordpress.com/2014/02/21/polymath8b-ix-large-quadratic-programs/#comment-297456 Nielsen]) | |||
| 246?# ([http://math.mit.edu/~primegaps/tuples/admissible_50_246.txt Clark-Jarvis]) | |||
| A 2-week computer calculation! | |||
|- | |||
| Apr 17 | |||
| | |||
| [http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m2_35410.mpl 35,410] [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie]) | |||
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m3_1649821.mpl 1,649,821] [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie]) | |||
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m4_75845707.mpl 75,845,707] [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie]) | |||
[http://www.cs.cmu.edu/~xfxie/project/admissible/k0/sol_varpi600d7m5_3473955908.mpl 3,473,955,908] [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302031 xfxie]) | |||
|398,646? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland]) | |||
25,816,462? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland]) | |||
1,541,858,666? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland]) | |||
84,449,123,072? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-302101 Sutherland]) | |||
| Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones | |||
|- | |||
| Apr 18 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/admissible_35410_398244.txt 398,244]? [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland]) | |||
[http://math.mit.edu/~drew/admissible_1649821_24798306.txt 24,798,306]? [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland]) | |||
1,541,183,756? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland]) | |||
84,449,103,908? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-303059 Sutherland]) | |||
| | |||
|- | |||
| Apr 28 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/admissible_35410_398130.txt 398,130] [m=2]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland]) | |||
[http://math.mit.edu/~drew/admissible_1649821_24797814.txt 24,797,814] [m=3]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland]) | |||
1,526,698,470? [m=4]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland]) | |||
83,833,839,882? [m=5]* ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-316813 Sutherland]) | |||
|- | |||
| May 1 | |||
| | |||
| | |||
| 81,973,172,502? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland]) | |||
2,165,674,446#? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland]) | |||
130,235,143,908#? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-319900 Sutherland]) | |||
| faster admissibility testing | |||
|- | |||
| May 3 | |||
| | |||
| | |||
| 1,460,493,420? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
80,088,836,006? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
1,488,227,220?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
81,912,638,914?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
2,111,605,786?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
127,277,395,046?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-321171 Sutherland]) | |||
| Fast admissibility testing for Hensley-Richards tuples | |||
|- | |||
| May 3 | |||
| | |||
| 3,393,468,735? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey]) | |||
2,113,163?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey]) | |||
105,754,479?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey]) | |||
5,274,206,963?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-322560 de Grey]) | |||
| | |||
| Improved hillclimbing; also confirmation of previous k values | |||
|- | |||
| May 4 | |||
| | |||
| | |||
| 79,929,339,154? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland]) | |||
[http://math.mit.edu/~drew/admissible_2113163_32588668.txt 32,588,668]?#* [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland]) | |||
2,111,597,632?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland]) | |||
126,630,432,986?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-323235 Sutherland]) | |||
|- | |||
| May 5 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/admissible_2113163_32285928.txt 32,285,928]?# [m=3] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-324263 Sutherland]) | |||
|- | |||
| May 9 | |||
| | |||
| | |||
| 1,460,485,532? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
79,929,332,990? [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
1,488,222,198?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
81,912,604,302?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
2,111,417,340?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
126,630,386,774?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-330204 Sutherland]) | |||
| Fast admissibility testing for Hensley-Richards sequences | |||
|- | |||
| May 14 | |||
| | |||
| | |||
| 1,440,495,268? [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
78,807,316,822 [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
1,467,584,468?* [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
80,761,835,464?* [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
2,082,729,956?# [m=4] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
124,840,189,042?# [m=5] ([http://terrytao.wordpress.com/2014/04/14/polymath8b-x-writing-the-paper-and-chasing-down-loose-ends/#comment-339197 Sutherland]) | |||
| Fast admissibility testing for Schinzel sequences | |||
|- | |||
| May 18 | |||
| | |||
| | |||
| 1,435,011,318? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland]) | |||
1,462,568,450?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland]) | |||
2,075,186,584?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-345117 Sutherland]) | |||
| Faster modified Schinzel sieve testing | |||
|- | |||
| May 23 | |||
| | |||
| | |||
| 1,424,944,070? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland]) | |||
1,452,348,402?* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-351013 Sutherland]) | |||
| Fast restricted greedy sieving | |||
|- | |||
| May 28 | |||
| | |||
| 52? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355568 de Grey]) | |||
51? [m=2] [GEH] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-355656 de Grey]) | |||
| 254? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_52_254.txt Clark-Jarvis]) | |||
252? [m=2] [GEH] ([http://math.mit.edu/~primegaps/tuples/admissible_51_252.txt Clark-Jarvis]) | |||
| New bounds for <math>M_{k,1/(k-1)}</math> | |||
|- | |||
| May 30 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/greedy_74487363_1404556152.txt 1,404,556,152]? [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland]) | |||
[http://math.mit.edu/~drew/greedy_75845707_1431556072.txt 1,431,556,072]* [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland]) | |||
[http://math.mit.edu/~drew/greedy_105754837_2031558336.txt 2,031,558,336]?# [m=4] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-357073 Sutherland]) | |||
| Heuristically determined shift for the shifted greedy sieve | |||
|- | |||
| June 8 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/schinzel_3473955908_80550202480.txt 80,550,202,480]* [m=5] ([http://terrytao.wordpress.com/2014/05/17/polymath-8b-xi-finishing-up-the-paper/#comment-366807 Sutherland]) | |||
| Verification of several previous bounds | |||
|- | |||
| June 23 | |||
| | |||
| | |||
| [http://math.mit.edu/~drew/schinzel_3393468735_78602310160.txt 78,602,310,160]? [m=5] ([http://terrytao.wordpress.com/2014/06/19/polymath8-wrapping-up/#comment-378098 Sutherland]) | |||
|} | |} | ||
| Line 759: | Line 1,054: | ||
# ? - unconfirmed or conditional | # ? - unconfirmed or conditional | ||
# ?? - theoretical limit of an analysis, rather than a claimed record | # ?? - theoretical limit of an analysis, rather than a claimed record | ||
# <nowiki>*</nowiki> - is majorized by an earlier but independent result | # <nowiki>*</nowiki> - is majorized by an earlier but independent or conditional result | ||
# <nowiki>#</nowiki> - bound does not rely on Deligne's theorems | # <nowiki>#</nowiki> - bound does not rely on Deligne's theorems | ||
# [EH] - bound is conditional the Elliott-Halberstam conjecture | # [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 | # [m=N] - bound on intervals containing N+1 consecutive primes, rather than two | ||
# strikethrough - values relied on a computation that has now been retracted | # 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>. | 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>. | ||
Latest revision as of 14:38, 5 July 2015
| Date | [math]\displaystyle{ \varpi }[/math] or [math]\displaystyle{ (\varpi,\delta) }[/math] | [math]\displaystyle{ k_0 }[/math] | [math]\displaystyle{ H }[/math] | Comments |
|---|---|---|---|---|
| Aug 10 2005 | 6 [EH] | 16 [EH] ([Goldston-Pintz-Yildirim]) | First bounded prime gap result (conditional on Elliott-Halberstam) | |
| May 14 2013 | 1/1,168 (Zhang) | 3,500,000 (Zhang) | 70,000,000 (Zhang) | All subsequent work (until the work of Maynard) is based on Zhang's breakthrough paper. |
| May 21 | 63,374,611 (Lewko) | Optimises Zhang's condition [math]\displaystyle{ \pi(H)-\pi(k_0) \gt k_0 }[/math]; can be reduced by 1 by parity considerations | ||
| May 28 | 59,874,594 (Trudgian) | Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] with [math]\displaystyle{ p_{m+1} \gt k_0 }[/math] | ||
| May 30 | 59,470,640 (Morrison)
58,885,998? (Tao) 59,093,364 (Morrison) 57,554,086 (Morrison) |
Uses [math]\displaystyle{ (p_{m+1},\ldots,p_{m+k_0}) }[/math] and then [math]\displaystyle{ (\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 (Morrison)
2,618,607 (Morrison) |
48,112,378 (Morrison)
42,543,038 (Morrison) 42,342,946 (Morrison) |
Optimizes Zhang's condition [math]\displaystyle{ \omega\gt 0 }[/math], and then uses an improved bound on [math]\displaystyle{ \delta_2 }[/math] | |
| Jun 1 | 42,342,924 (Tao) | Tiny improvement using the parity of [math]\displaystyle{ k_0 }[/math] | ||
| Jun 2 | 866,605 (Morrison) | 13,008,612 (Morrison) | Uses a further improvement on the quantity [math]\displaystyle{ \Sigma_2 }[/math] in Zhang's analysis (replacing the previous bounds on [math]\displaystyle{ \delta_2 }[/math]) | |
| Jun 3 | 1/1,040? (v08ltu) | 341,640 (Morrison) | 4,982,086 (Morrison)
4,802,222 (Morrison) |
Uses a different method to establish [math]\displaystyle{ DHL[k_0,2] }[/math] that removes most of the inefficiency from Zhang's method. |
| Jun 4 | 1/224?? (v08ltu)
1/240?? (v08ltu) |
4,801,744 (Sutherland)
4,788,240 (Sutherland) |
Uses asymmetric version of the Hensley-Richards tuples | |
| Jun 5 | 34,429? (Paldi/v08ltu) | 4,725,021 (Elsholtz)
4,717,560 (Sutherland) 397,110? (Sutherland) 4,656,298 (Sutherland) 389,922 (Sutherland) 388,310 (Sutherland) 388,284 (Castryck) 388,248 (Sutherland) 387,982 (Castryck) 387,974 (Castryck) |
[math]\displaystyle{ 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]\displaystyle{ H }[/math] bound obtained by a hybrid Schinzel/greedy (or "greedy-greedy") sieve | |
| Jun 6 |
|
387,960 (Angelveit)
387,904 (Angeltveit)
|
Improved [math]\displaystyle{ H }[/math]-bounds based on experimentation with different residue classes and different intervals, and randomized tie-breaking in the greedy sieve. | |
| Jun 7 |
[math]\displaystyle{ 828 \varpi + 172\delta \lt 1 }[/math] (v08ltu/Green) |
26,024? (vo8ltu) |
387,534 (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 | 286,224 (Sutherland)
285,752 (pedant-Sutherland) |
values of [math]\displaystyle{ \varpi,\delta,k_0 }[/math] now confirmed; most tuples available on dropbox. New bounds on [math]\displaystyle{ H }[/math] obtained via iterated merging using a randomized greedy sieve. | ||
| Jun 9 | 181,000*? (Pintz) | 2,530,338*? (Pintz) | New bounds on [math]\displaystyle{ H }[/math] obtained by interleaving iterated merging with local optimizations. | |
| Jun 10 | 23,283? (Harcos/v08ltu) | 285,210 (Sutherland) | More efficient control of the [math]\displaystyle{ \kappa }[/math] error using the fact that numbers with no small prime factor are usually coprime | |
| Jun 11 | 252,804 (Sutherland) | More refined local "adjustment" optimizations, as detailed here.
An issue with the [math]\displaystyle{ k_0 }[/math] computation has been discovered, but is in the process of being repaired. | ||
| Jun 12 | 22,951 (Tao/v08ltu)
22,949 (Harcos) |
249,180 (Castryck) | Improved bound on [math]\displaystyle{ k_0 }[/math] avoids the technical issue in previous computations. | |
| Jun 13 | ||||
| Jun 14 | 248,898 (Sutherland) | |||
| Jun 15 | [math]\displaystyle{ 348\varpi+68\delta \lt 1 }[/math]? (Tao) | 6,330? (v08ltu)
6,329? (Harcos) 6,329 (v08ltu) |
60,830? (Sutherland) | Taking more advantage of the [math]\displaystyle{ \alpha }[/math] convolution in the Type III sums |
| Jun 16 | [math]\displaystyle{ 348\varpi+68\delta \lt 1 }[/math] (v08ltu)
|
60,760* (Sutherland)
|
Attempting to make the Weyl differencing more efficient; unfortunately, it did not work | |
| Jun 18 | 5,937? (Pintz/Tao/v08ltu)
5,672? (v08ltu) 5,459? (v08ltu) 5,454? (v08ltu) 5,453? (v08ltu) |
60,740 (xfxie)
58,866? (Sun) 53,898? (Sun) 53,842? (Sun) |
A new truncated sieve of Pintz virtually eliminates the influence of [math]\displaystyle{ \delta }[/math] | |
| Jun 19 | 5,455? (v08ltu)
5,453? (v08ltu) 5,452? (v08ltu) |
53,774? (Sun)
53,672*? (Sun) |
Some typos in [math]\displaystyle{ \kappa_3 }[/math] estimation had placed the 5,454 and 5,453 values of [math]\displaystyle{ k_0 }[/math] into doubt; however other refinements have counteracted this | |
| Jun 20 | [math]\displaystyle{ 178\varpi + 52\delta \lt 1 }[/math]? (Tao)
[math]\displaystyle{ 148\varpi + 33\delta \lt 1 }[/math]? (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]\displaystyle{ 148\varpi + 33\delta \lt 1 }[/math] (v08ltu) | 1,470 (v08ltu)
1,467 (v08ltu) |
12,042 (Engelsma) | Systematic tables of tuples of small length have been set up here and here (update: As of June 27 these tables have been merged and uploaded to an online database of current bounds on [math]\displaystyle{ H(k) }[/math] for [math]\displaystyle{ k }[/math] up to 5000). |
| Jun 22 | Slight improvement in the [math]\displaystyle{ \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]\displaystyle{ k_0 }[/math] | |||
| Jun 23 | 1,466 (Paldi/Harcos) | 12,006 (Engelsma) | An improved monotonicity formula for [math]\displaystyle{ G_{k_0-1,\tilde \theta} }[/math] reduces [math]\displaystyle{ \kappa_3 }[/math] somewhat | |
| Jun 24 | [math]\displaystyle{ (134 + \tfrac{2}{3}) \varpi + 28\delta \le 1 }[/math]? (v08ltu)
[math]\displaystyle{ 140\varpi + 32 \delta \lt 1 }[/math]? (Tao)
|
1,268? (v08ltu) | 10,206? (Engelsma) | A theoretical gain from rebalancing the exponents in the Type I exponential sum estimates |
| Jun 25 | [math]\displaystyle{ 116\varpi+30\delta\lt 1 }[/math]? (Fouvry-Kowalski-Michel-Nelson/Tao) | 1,346? (Hannes)
1,007? (Hannes) |
10,876? (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]\displaystyle{ 116\varpi + 25.5 \delta \lt 1 }[/math]? (Nielsen)
[math]\displaystyle{ (112 + \tfrac{4}{7}) \varpi + (27 + \tfrac{6}{7}) \delta \lt 1 }[/math]? (Tao) |
962? (Hannes) | 7,470? (Engelsma) | Beginning to flesh out various "levels" of Type I, Type II, and Type III estimates, see this page, in particular optimising van der Corput in the Type I sums. Integrated tuples page now online. |
| Jun 27 | [math]\displaystyle{ 108\varpi + 30 \delta \lt 1 }[/math]? (Tao) | 902? (Hannes) | 6,966? (Engelsma) | Improved the Type III estimates by averaging in [math]\displaystyle{ \alpha }[/math]; also some slight improvements to the Type II sums. Tuples page is now accepting submissions. |
| Jul 1 | [math]\displaystyle{ (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1 }[/math]? (Tao) |
873? (Hannes)
|
Refactored the final Cauchy-Schwarz in the Type I sums to rebalance the off-diagonal and diagonal contributions | |
| Jul 5 | [math]\displaystyle{ (93 + \frac{1}{3}) \varpi + (26 + \frac{2}{3}) \delta \lt 1 }[/math] (Tao) |
Weakened the assumption of [math]\displaystyle{ x^\delta }[/math]-smoothness of the original moduli to that of double [math]\displaystyle{ x^\delta }[/math]-dense divisibility | ||
| Jul 10 | 7/600? (Tao) | An in principle refinement of the van der Corput estimate based on exploiting additional averaging | ||
| Jul 19 | [math]\displaystyle{ (85 + \frac{5}{7})\varpi + (25 + \frac{5}{7}) \delta \lt 1 }[/math]? (Tao) | A more detailed computation of the Jul 10 refinement | ||
| Jul 20 | Jul 5 computations now confirmed | |||
| Jul 27 | 633 (Tao)
632 (Harcos) |
4,686 (Engelsma) | ||
| Jul 30 | [math]\displaystyle{ 168\varpi + 48\delta \lt 1 }[/math]# (Tao) | 1,788# (Tao) | 14,994# (Sutherland) | Bound obtained without using Deligne's theorems. |
| Aug 17 | 1,783# (xfxie) | 14,950# (Sutherland) | ||
| Oct 3 | 13/1080?? (Nelson/Michel/Tao) | 604?? (Tao) | 4,428?? (Engelsma) | Found an additional variable to apply van der Corput to |
| Oct 11 | [math]\displaystyle{ 83\frac{1}{13}\varpi + 25\frac{5}{13} \delta \lt 1 }[/math]? (Tao) | 603? (xfxie) | 4,422?(Engelsma)
12 [EH] (Maynard) |
Worked out the dependence on [math]\displaystyle{ \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) | Announced at a talk in Oberwolfach | ||
| Oct 24 | 110#? (Maynard) | 628#? (Clark-Jarvis) | With this value of [math]\displaystyle{ k_0 }[/math], the value of [math]\displaystyle{ H }[/math] given is best possible (and similarly for smaller values of [math]\displaystyle{ k_0 }[/math]) | |
| Nov 19 | 105# (Maynard)
5 [EH] (Maynard) |
600# (Maynard/Clark-Jarvis) | One also gets three primes in intervals of length 600 if one assumes Elliott-Halberstam | |
| Nov 20 |
|
|
Optimizing the numerology in Maynard's large k analysis; unfortunately there was an error in the variance calculation | |
| Nov 21 | 68?? (Maynard)
582#*? (Nielsen]) 59,451 [m=2]#? (Nielsen]) 42,392 [m=2]? (Nielsen) |
356?? (Clark-Jarvis) | Optimistically inserting the Polymath8a distribution estimate into Maynard's low k calculations, ignoring the role of delta | |
| Nov 22 | 388*? (xfxie)
448#*? (Nielsen) 43,134 [m=2]#? (Nielsen) |
698,288 [m=2]#? (Sutherland)
484,290 [m=2]? (Sutherland) 484,276 [m=2]? (Sutherland) |
Uses the m=2 values of k_0 from Nov 21 | |
| Nov 23 | 493,528 [m=2]#? Sutherland
493,510 [m=2]#? Sutherland 484,260 [m=2]? (Sutherland) 493,458 [m=2]#? Sutherland |
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| Nov 24 | 484,234 [m=2]? (Sutherland)
493,442 [m=2]#? (Sutherland) 484,192 [m=2]? (Sutherland) |
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| Nov 25 | 385#*? (xfxie) | 484,176 [m=2]? (Sutherland)
493,436[m=2]#? (Sutherland) |
Using the exponential moment method to control errors | |
| Nov 26 | 102# (Nielsen) | 493,426 [m=2]#? (Sutherland)
576# (Clark-Jarvis) |
Optimising the original Maynard variational problem | |
| Nov 27 | 484,162 [m=2]? (Sutherland)
484,142 [m=2]? (Sutherland) |
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| Nov 28 | 484,136 [m=2]? (Sutherland
484,126 [m=2]? (Sutherland) |
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| Dec 4 | 64#? (Nielsen) | 330#? (Clark-Jarvis) | Searching over a wider range of polynomials than in Maynard's paper | |
| Dec 6 | 493,408 [m=2]#? (Sutherland) | |||
| Dec 19 | 59#? (Nielsen)
10,000,000? [m=3] (Tao) 1,700,000? [m=3] (Tao) 38,000? [m=2] (Tao) |
300#? (Clark-Jarvis)
182,087,080? [m=3] (Sutherland) 179,933,380? [m=3] (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]\displaystyle{ M_k \geq \frac{k}{k-1} \log k - O(1) }[/math] lower bound. | |
| Dec 20 | 55#? (Nielsen) 36,000? [m=2] (xfxie) |
175,225,874? [m=3] (Sutherland)
27,398,976? [m=3] (Sutherland) 26,682,014? [m=3] (Sutherland) 431,682? [m=2] (Sutherland) 430,448? [m=2] (Sutherland) 429,822? [m=2] (Sutherland)
272#? (Clark-Jarvis) |
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| Dec 21 | 1,640,042? [m=3] (Sutherland)
1,631,027? [m=3] (Sutherland)
|
429,798? [m=2] (Sutherland)
25,602,438? [m=3] (Sutherland) 405,528? [m=2] (Sutherland)
25,533,684? [m=3] (Sutherland) 395,264? [m=2] (Sutherland) 395,178? [m=2] (Sutherland) 25,527,718? [m=3] (Sutherland)
24,490,758? [m=3] (Sutherland) |
Optimising the explicit lower bound [math]\displaystyle{ M_k \geq \log k-O(1) }[/math] | |
| Dec 22 | 1,628,944? [m=3] (Castryck)
75,000,000? [m=4] (Castryck) 3,400,000,000? [m=5] (Castryck) 5,511 [EH] [m=3] (Sutherland) 2,114,964#? [m=3] (Sutherland) 309,954? [EH] [m=5] (Sutherland) 74,487,363? [m=4] (xfxie) |
395,154? [m=2] (Sutherland)
24,490,410? [m=3] (Sutherland)
395,122? [m=2] (Sutherland)
1,523,781,850? [m=4] (Sutherland) 82,575,303,678? [m=5] (Sutherland) 52,130? [EH] [m=3] (Sutherland) 33,661,442?# [m=3] (Sutherland) 24,462,790? [m=3] (Sutherland) 4,316,446? [EH] [m=5] (Sutherland) |
A numerical precision issue was discovered in the earlier m=4 calculations | |
| Dec 23 | 41,589? [EH] [m=4] (Sutherland)
105,754,838#? [m=4] (Sutherland) 5,300,000,000#? [m=5] (Sutherland) |
24,462,774? [m=3] (Sutherland)
1,512,832,950? [m=4] (Sutherland) 4,146,936? [EH] [m=5] (Sutherland) 52,116 [EH] [m=3] (Sutherland) 474,600? [EH] [m=4] (Sutherland) 474,460? [EH] [m=4] (Sutherland) 4,143,140? [EH] [m=5] (Sutherland) 32,313,942#? [m=3] (Sutherland) 2,186,561,568#? [m=4] (Sutherland) 474,372? [EH] [m=4] (Sutherland) 131,161,149,090#? [m=5] (Sutherland) |
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| Dec 24 | 474,320? [EH] [m=4] (Sutherland)
4,137,872? [EH] [m=5] (Sutherland) 24,462,654? [m=3] (Sutherland) 1,497,901,734? [m=4] (Sutherland) 32,313,878#? [m=3] (Sutherland) |
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| Dec 28 | 474,296? [EH] [m=4] (Sutherland)
4,137,854 [EH] [m=5] (Sutherland) |
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| Jan 2 2014 | 474,290? [EH] [m=4] (Sutherland) | |||
| Jan 6 | 54# (Nielsen) | 270# (Clark-Jarvis) | ||
| Jan 8 | 4 [GEH] (Nielsen) | 8 [GEH] (Nielsen) | Using a "gracefully degrading" lower bound for the numerator of the optimisation problem. Calculations confirmed here. | |
| Jan 9 | 474,266 [EH] [m=4] (Sutherland) | |||
| Jan 28 | 395,106? [m=2] (Sutherland) | |||
| Jan 29 | 3 [GEH] (Nielsen) | 6 [GEH] (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 here | |||
| Feb 17 | 53?# (Nielsen) | 264?# (Clark-Jarvis) | Managed to get the epsilon trick to be computationally feasible for medium k | |
| Feb 22 | 51?# (Nielsen) | 252?# (Clark-Jarvis) | More efficient matrix computation allows for higher degrees to be used | |
| Mar 4 | Jan 6 computations confirmed | |||
| Apr 14 | 50?# (Nielsen) | 246?# (Clark-Jarvis) | A 2-week computer calculation! | |
| Apr 17 | 35,410 [m=2]* (xfxie)
75,845,707 [m=4]* (xfxie) 3,473,955,908 [m=5]* (xfxie) |
398,646? [m=2]* (Sutherland)
25,816,462? [m=3]* (Sutherland) 1,541,858,666? [m=4]* (Sutherland) 84,449,123,072? [m=5]* (Sutherland) |
Redoing the m=2,3,4,5 computations using the confirmed MPZ estimates rather than the unconfirmed ones | |
| Apr 18 | 398,244? [m=2]* (Sutherland)
24,798,306? [m=3]* (Sutherland) 1,541,183,756? [m=4]* (Sutherland) 84,449,103,908? [m=5]* (Sutherland) |
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| Apr 28 | 398,130 [m=2]* (Sutherland)
24,797,814 [m=3]* (Sutherland) 1,526,698,470? [m=4]* (Sutherland) 83,833,839,882? [m=5]* (Sutherland) | |||
| May 1 | 81,973,172,502? [m=5] (Sutherland)
2,165,674,446#? [m=4] (Sutherland) 130,235,143,908#? [m=5] (Sutherland) |
faster admissibility testing | ||
| May 3 | 1,460,493,420? [m=4] (Sutherland)
80,088,836,006? [m=5] (Sutherland) 1,488,227,220?* [m=4] (Sutherland) 81,912,638,914?* [m=5] (Sutherland) 2,111,605,786?# [m=4] (Sutherland) 127,277,395,046?# [m=5] (Sutherland) |
Fast admissibility testing for Hensley-Richards tuples | ||
| May 3 | 3,393,468,735? [m=5] (de Grey)
2,113,163?# [m=3] (de Grey) 105,754,479?# [m=4] (de Grey) 5,274,206,963?# [m=5] (de Grey) |
Improved hillclimbing; also confirmation of previous k values | ||
| May 4 | 79,929,339,154? [m=5] (Sutherland)
32,588,668?#* [m=3] (Sutherland) 2,111,597,632?# [m=4] (Sutherland) 126,630,432,986?# [m=5] (Sutherland) | |||
| May 5 | 32,285,928?# [m=3] (Sutherland) | |||
| May 9 | 1,460,485,532? [m=4] (Sutherland)
79,929,332,990? [m=5] (Sutherland) 1,488,222,198?* [m=4] (Sutherland) 81,912,604,302?* [m=5] (Sutherland) 2,111,417,340?# [m=4] (Sutherland) 126,630,386,774?# [m=5] (Sutherland) |
Fast admissibility testing for Hensley-Richards sequences | ||
| May 14 | 1,440,495,268? [m=4] (Sutherland)
78,807,316,822 [m=5] (Sutherland) 1,467,584,468?* [m=4] (Sutherland) 80,761,835,464?* [m=5] (Sutherland) 2,082,729,956?# [m=4] (Sutherland) 124,840,189,042?# [m=5] (Sutherland) |
Fast admissibility testing for Schinzel sequences | ||
| May 18 | 1,435,011,318? [m=4] (Sutherland)
1,462,568,450?* [m=4] (Sutherland) 2,075,186,584?# [m=4] (Sutherland) |
Faster modified Schinzel sieve testing | ||
| May 23 | 1,424,944,070? [m=4] (Sutherland)
1,452,348,402?* [m=4] (Sutherland) |
Fast restricted greedy sieving | ||
| May 28 | 52? [m=2] [GEH] (de Grey)
51? [m=2] [GEH] (de Grey) |
254? [m=2] [GEH] (Clark-Jarvis)
252? [m=2] [GEH] (Clark-Jarvis) |
New bounds for [math]\displaystyle{ M_{k,1/(k-1)} }[/math] | |
| May 30 | 1,404,556,152? [m=4] (Sutherland)
1,431,556,072* [m=4] (Sutherland) 2,031,558,336?# [m=4] (Sutherland) |
Heuristically determined shift for the shifted greedy sieve | ||
| June 8 | 80,550,202,480* [m=5] (Sutherland) | Verification of several previous bounds | ||
| June 23 | 78,602,310,160? [m=5] (Sutherland) |
Legend:
- ? - unconfirmed or conditional
- ?? - theoretical limit of an analysis, rather than a claimed record
- * - is majorized by an earlier but independent or conditional result
- # - 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]\displaystyle{ H }[/math] for various key values of [math]\displaystyle{ k_0 }[/math].
