Problem Statement
Let $k\geq 2$ and $q(n,k)$ denote the least prime which does not divide $\prod_{1\leq i\leq k}(n+i)$. Is it true that, if $k$ is fixed and $n$ is sufficiently large, we have\[q(n,k)<(1+o(1))\log n?\]
Categories:
Number Theory
Progress
A problem of Erdős and Pomerance.The bound $q(n,k)<(1+o(1))k\log n$ is easy. It may be true this improved bound holds even up to $k=o(\log n)$.
A heuristic argument in favour of this is provided by Tao in the comments.
See also [457].
Source: erdosproblems.com/663 | Last verified: January 16, 2026