The work with the Polymath project was published on Research in the Mathematical Sciences.

- D. H. J. Polymath, et al. Variants of the Selberg sieve, and bounded intervals containing many primes.
*Research in the Mathematical Sciences*, 2014, 1(12): 1-83. [DOI]

Here is the list of Polymath8b authors (arranged in alphabetical order of surname): Ignace Bogaert, Aubrey de Grey, Gergely Harcos, Emmanuel Kowalski, Philippe Michel, James Maynard, Paul Nelson, Pace Nielsen, Eytan Paldi, Andrew V. Sutherland, Terence Tao, Xiao-Feng Xie

*Abstract*: For any $latex m \geq 1$, let $latex H_m$ denote the quantity $latex \liminf_{n \to \infty} (p_{n+m}-p_n)$, where $latex p_n$ is the $latex n^{\text{th}}$ prime. A celebrated recent result of Zhang showed the finiteness of $latex H_1$, with the explicit bound $latex H_1 \leq 70000000$. This was then improved by us (the Polymath8 project) to $latex H_1 \leq 4680$, and then by Maynard to $latex H_1 \leq 600$, who also established for the first time a finiteness result for $latex H_m$ for $latex m \geq 2$, and specifically that $latex H_m \ll m^3 e^{4m}$. If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound $latex H_1 \leq 12$, improving upon the previous bound $latex H_1 \leq 16$ of Goldston, Pintz, and Y{\i}ld{\i}r{\i}m, as well as the bound $latex H_m \ll m^3 e^{2m}$. In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further, and by performing more extensive numerical calculations. As a consequence, we can obtain the bound $latex H_1 \leq 246$ unconditionally, and $latex H_1 \leq 6$ under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple $latex (h_1,h_2,h_3)$, there are infinitely many $latex n$ for which at least two of $latex n+h_1,n+h_2,n+h_3$ are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds, or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most $latex 2$, or both. We also modify the “parity problem” argument of Selberg to show that the $latex H_1 \leq 6$ bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger $latex m$, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound $latex H_m \ll m e^{(4-\frac{28}{157})m}$, or $latex H_m \ll m e^{2m}$ under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for $latex H_m$ when $latex m=2,3,4,5$.