A problem of Erdős and Pomerance.


More generally, let $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. This problem asks whether $q(n,\log n)\geq (2+\epsilon)\log n$ infinitely often. Taking $n$ to be the product of primes between $\log n$ and $(2+o(1))\log n$ gives an example where\[q(n,\log n)\geq (2+o(1))\log n.\]Can one prove that $q(n,\log n)<(1-\epsilon)(\log n)^2$ for all large $n$ and some $\epsilon>0$?


See also [663].

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This page was last edited 07 October 2025.