Frangible prime gaps

Abstract

A prime gap $(p_n, p_{n+1})$ is \textit{frangible} if the smallest prime divisor of every integer $m$ with $p_n<m<p_{n+1}$ is strictly less than the size of the gap, $p_{n+1}-p_n$. Gafni and Tao recently proved that the number of frangible gaps up to $X$ is $\ll X/\log^2 X$, answering a question of Erdős. They also showed that under the Dickson--Hardy--Littlewood prime tuples conjecture this number is $\sim c X/\log^2 X$ for a constant $c$, for which they obtained a rough estimate. We describe improved methods for estimating this constant under the same hypothesis, and establish that its value lies in $[2.7416, 2.7984]$, confirming a speculation of Gafni and Tao. In addition, we present the results of a complete survey of frangible gaps up to $10^{17}$. An ensuing empirical measurement of $c$ matches our estimate very well.