Rough numbers between consecutive primes

Abstract

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap, confirming a prediction of Erd\H{o}s. In fact the number $N(X)$ of exceptional gaps with $p_n \in [X,2X]$ is shown to be at most $O(X/\log^2 X)$. Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic $N(X) \sim c X / \log^2 X$ for an explicit constant $c>0$, which we believe to be between $2.7$ and $2.8$. To obtain our results in their full strength we rely on the asymptotics for singular series developed by Montgomery and Soundararajan.