Using a sieve-theoretic argument, we show that almost all gaps (pₙ, p₍+₁) between consecutive primes pₙ, p₍+₁ contain a natural number m whose least prime factor p (m) is at least the length p₍+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/² X). Assuming a form of the Hardy--Littlewood prime tuples conjecture, we establish a more precise asymptotic N (X) c X / 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.
Gafni et al. (Fri,) studied this question.