Problem Statement
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide\[\prod_{1\leq i\leq \log n}(n+i)?\]
Categories:
Number Theory
Progress
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].
Source: erdosproblems.com/457 | Last verified: January 15, 2026