OPEN
This is open, and cannot be resolved with a finite computation.
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\[q(n,k)<(1+o(1))\log n?\]
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].
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This page was last edited 02 December 2025.
Additional thanks to: Terence Tao
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #663, https://www.erdosproblems.com/663, accessed 2026-01-14
