From Polymath Wiki

This is the home page for the Polymath8 project, which has two components:

Polymath8a, "Bounded gaps between primes", was a project to improve the bound H=H_1 on the least gap between consecutive primes that was attained infinitely often, by developing the techniques of Zhang. This project concluded with a bound of H = 4,680.

Polymath8b, "Bounded intervals with many primes", was project to improve the value of H_1 further, as well as H_m (the least gap between primes with m-1 primes between them that is attained infinitely often), by combining the Polymath8a results with the techniques of Maynard. This project concluded with a bound of H=246, as well as additional bounds on H_m (see below).

World records

Current records

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

[math]m[/math] Conjectural Assuming EH Without EH Without EH or Deligne 1 2 6 (on GEH) 12 [M] (on EH only) 246 246 2 6 252 (on GEH) 270 (on EH only) 395,106 474,266 3 8 52,116 24,462,654 32,285,928 4 12 474,266 1,404,556,152 2,031,558,336 5 16 4,137,854 78,602,310,160 124,840,189,042 [math]m[/math] [math]\displaystyle (1+o(1)) m \log m[/math] [math]\displaystyle O( m e^{2m} )[/math] [math]O( \exp( 3.815 m) ) [BI][/math] [math]O( m \exp((4 - \frac{4}{43}) m) )[/math]

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

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

[math]H = H_1[/math] is a quantity such that there are infinitely many pairs of consecutive primes of distance at most [math]H[/math] apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project).

is a quantity such that there are infinitely many pairs of consecutive primes of distance at most apart. Would like to be as small as possible (this is a primary goal of the Polymath8 project). [math]k_0[/math] is a quantity such that every admissible [math]k_0[/math] -tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in [math]k_0[/math] lead to improvements in [math]H[/math] . (The relationship is roughly of the form [math]H \sim k_0 \log k_0[/math] ; see the page on finding narrow admissible tuples.) More recent improvements on [math]k_0[/math] have come from solving a Selberg sieve variational problem.

is a quantity such that every admissible -tuple has infinitely many translates which each contain at least two primes. Would like to be as small as possible. Improvements in lead to improvements in . (The relationship is roughly of the form ; see the page on finding narrow admissible tuples.) More recent improvements on have come from solving a Selberg sieve variational problem. [math]\varpi[/math] is a technical parameter related to a specialized form of the Elliott-Halberstam conjecture. Would like to be as large as possible. Improvements in [math]\varpi[/math] lead to improvements in [math]k_0[/math] , as described in the page on Dickson-Hardy-Littlewood theorems. In more recent work, the single parameter [math]\varpi[/math] is replaced by a pair [math](\varpi,\delta)[/math] (in previous work we had [math]\delta=\varpi[/math] ). These estimates are obtained in turn from Type I, Type II, and Type III estimates, as described at the page on distribution of primes in smooth moduli.

Timeline of bounds

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

Polymath threads

Writeup

Here are the Polymath8 grant acknowledgments.

Code and data

Tuples applet

Here is a small javascript applet that illustrates the process of sieving for an admissible 632-tuple of diameter 4680 (the sieved residue classes match the example in the paper when translated to [0,4680]).

The same applet can also be used to interactively create new admissible tuples. The default sieve interval is [0,400], but you can change the diameter to any value you like by adding “?d=nnnn” to the URL (e.g. use https://math.mit.edu/~primegaps/sieve.html?d=4680 for diameter 4680). The applet will highlight a suggested residue class to sieve in green (corresponding to a greedy choice that doesn’t hit the end points), but you can sieve any classes you like.

You can also create sieving demos similar to the k=632 example above by specifying a list of residues to sieve. A longer but equivalent version of the “?ktuple=632″ URL is

https://math.mit.edu/~primegaps/sieve.html?d=4680&r=1,1,4,3,2,8,2,14,13,8,29,31,33,28,6,49,21,47,58,35,57,44,1,55,50,57,9,91,87,45,89,50,16,19,122,114,151,66

The numbers listed after “r=” are residue classes modulo increasing primes 2,3,5,7,…, omitting any classes that do not require sieving — the applet will automatically skip such primes (e.g. it skips 151 in the k=632 example).

A few caveats: You will want to run this on a PC with a reasonably wide display (say 1360 or greater), and it has only been tested on the current versions of Chrome and Firefox.

Errata

Page numbers refer to the file linked to for the relevant paper.

Errata for Zhang's "Bounded gaps between primes" Page 5: In the first display, [math]\mathcal{E}[/math] should be multiplied by [math]\mathcal{L}^{2k_0+2l_0}[/math] , because [math]\lambda(n)^2[/math] in (2.2) can be that large, cf. (2.4). Page 14: In the final display, the constraint [math](n,d_1=1[/math] should be [math](n,d_1)=1[/math] . Page 35: In the display after (10.5), the subscript on [math]{\mathcal J}_i[/math] should be deleted. Page 36: In the third display, a factor of [math]\tau(q_0r)^{O(1)}[/math] may be needed on the right-hand side (but is ultimately harmless). Page 38: In the display after (10.14), [math]\xi(r,a;q_1,b_1;q_2,b_2;n,k)[/math] should be [math]\xi(r,a;k;q_1,b_1;q_2,b_2;n)[/math] . Page 42: In (12.3), [math]B[/math] should probably be 2. Page 47: In the third display after (13.13), the condition [math]l \in {\mathcal I}_i(h)[/math] should be [math]l \in {\mathcal I}_i(sh)[/math] . Page 49: In the top line, a comma in [math](h_1,h_2;,n_1,n_2)[/math] should be deleted. Page 51: In the penultimate display, one of the two consecutive commas should be deleted. Page 54: Three displays before (14.17), [math]\bar{r_2}(m_1+m_2)q[/math] should be [math]\bar{r_2}(m_1+m_2)/q[/math] . Errata for Motohashi-Pintz's "A smoothed GPY sieve", version 1. Update: the errata below have been corrected in the most recent arXiv version of the paper. Page 31: The estimation of (5.14) by (5.15) does not appear to be justified. In the text, it is claimed that the second summation in (5.14) can be treated by a variant of (4.15); however, whereas (5.14) contains a factor of [math](\log \frac{R}{|D|})^{2\ell+1}[/math] , (4.15) contains instead a factor of [math](\log \frac{R/w}{|K|})^{2\ell+1}[/math] which is significantly smaller (K in (4.15) plays a similar role to D in (5.14)). As such, the crucial gain of [math]\exp(-k\omega/3)[/math] in (4.15) does not seem to be available for estimating the second sum in (5.14). Errata for Pintz's "A note on bounded gaps between primes", version 1. Update: the errata below have been corrected in subsequent versions of Pintz's paper. Page 7: In (2.39), the exponent of [math]3a/2[/math] should instead be [math]-5a/2[/math] (it comes from dividing (2.38) by (2.37)). This impacts the numerics for the rest of the paper. Page 8: The "easy calculation" that the relative error caused by discarding all but the smooth moduli appears to be unjustified, as it relies on the treatment of (5.14) in Motohashi-Pintz which has the issue pointed out in 2.1 above.

Other relevant blog posts

MathOverflow

Wikipedia and other references

Recent papers and notes

Media

Bibliography

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