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

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 , let denote the quantity , where is the prime. A celebrated recent result of Zhang showed the finiteness of , with the explicit bound . This was then improved by us (the Polymath8 project) to , and then by Maynard to , who also established for the first time a finiteness result for for , and specifically that . If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound , improving upon the previous bound of Goldston, Pintz, and Y{\i}ld{\i}r{\i}m, as well as the bound . 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 unconditionally, and under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture we show the stronger statement that for any admissible triple , there are infinitely many for which at least two of 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 , or both. We also modify the “parity problem” argument of Selberg to show that the bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger , we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound , or under the assumption of the Elliott-Halberstam conjecture. We also obtain explicit upper bounds for when .