Annals of Mathematics
Pages 1121-1174 from Volume 179 (2014), Issue 3 by Yitang Zhang
It is proved that
where pn is the n-th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.