Timeline of prime gap bounds
| Date | [math]\displaystyle{ \varpi }[/math] or [math]\displaystyle{ (\varpi,\delta) }[/math] | [math]\displaystyle{ k_0 }[/math] | [math]\displaystyle{ H }[/math] | Comments |
|---|---|---|---|---|
| 10 Aug 2005 | 6 [EH] | 16 [EH] ([Goldston-Pintz-Yildirim]) | First bounded prime gap result (conditional on Elliott-Halberstam) | |
| 14 May 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. |
| 21 May | 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 | ||
| 28 May | 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] | ||
| 30 May | 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 | ||
| 31 May | 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] | |
| 1 Jun | 42,342,924 (Tao) | Tiny improvement using the parity of [math]\displaystyle{ k_0 }[/math] | ||
| 2 Jun | 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]) | |
| 3 Jun | 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. |
| 4 Jun | 1/224?? (v08ltu)
1/240?? (v08ltu) |
4,801,744 (Sutherland)
4,788,240 (Sutherland) |
Uses asymmetric version of the Hensley-Richards tuples | |
| 5 Jun | 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 | |
| 6 Jun |
|
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. | |
| 7 Jun |
[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 |
|||
| Nov 24 | 484,234 [m=2]? (Sutherland)
493,442 [m=2]#? (Sutherland) 484,192 [m=2]? (Sutherland) |
|||
| 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) |
|||
| 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) |
||
| 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 |
Legend:
- ? - unconfirmed or conditional
- ?? - theoretical limit of an analysis, rather than a claimed record
- * - is majorized by an earlier but independent result
- # - bound does not rely on Deligne's theorems
- [EH] - bound is conditional the Elliott-Halberstam conjecture
- [m=N] - bound on intervals containing N+1 consecutive primes, rather than two
- strikethrough - values relied on a computation that has now been retracted
See also the article on Finding narrow admissible tuples for benchmark values of [math]\displaystyle{ H }[/math] for various key values of [math]\displaystyle{ k_0 }[/math].
