Rough numbers between consecutive primes

Using a sieve-theoretic argument, we show that almost all gaps (pₙ, pₙ₊₁) between consecutive primes pₙ, pₙ₊₁ contain a natural number m whose least prime factor $p(m)$ is at least the length pₙ₊₁ - pₙ of the gap, confirming a prediction of Erdős. In fact the number $N(X)$ of exceptional gaps with pₙ ∈ [X,2X] is shown to be at most O(X/log² X). Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic N(X) ~ c X / log² 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.