Problem Statement
Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$?
Categories:
Number Theory Primes
Progress
A question of Erdős reported in problem A2 of Guy's collection [Gu04].Examples are $p=101$ and $p=211$. Erdős suggested it may be easier to show that there are infinitely many $n$ such that, if $l!<n\leq (l+1)!$, then all the prime factors of $n$ are $>l$, and all the numbers $n-k!$ are composite for $1\leq k\leq l$.
Source: erdosproblems.com/1059 | Last verified: January 19, 2026