Bounded gaps between primes

Abstract

It is proved that $$ \liminf_{n\to\infty}(p_{n+1}-p_n)<7\times 10^7, $$ where $p_n$ 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.