Large gaps between consecutive prime numbers


Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdős, we show that \[ G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},\] where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.


Kevin Ford

University of Illinois at Urbana-Champaign Urbana, IL

Ben Green

Mathematical Institute Oxford England

Sergei Konyagin

Steklov Mathematical Institute Moscow, Russia

Terence Tao

University of California Los Angeles CA