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\[q(n,\log n)\geq (2+o(1))\log n.\]GPT-5.2 Pro (prompted by Barreto) has given
a construction that gives an affirmative answer to the original question, with the constant $2$ replaced by $\frac{3}{\log 4}\approx 2.16$, and GPT later proved this with $2$ replaced by an arbitrarily large constant.
An elaboration sketched by Tao in the comments proves that there are infinitely many $n$ such that\[q(n,\log n) > \frac{1-o(1)}{2}\frac{\log\log n}{\log\log\log n}\log n.\]Tao suggests that standard heuristics suggest that this lower bound is also (up to constants) an upper bound for all $n$.
See also
[663], and
[1181] for upper bounds on $q(n,\log n)$.
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This page was last edited 07 March 2026. View history
Additional thanks to: Kevin Barreto and 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 #457, https://www.erdosproblems.com/457, accessed 2026-04-11
