Problem 1059

Erdős Problem #1059: declared status 'open'. Formalized: yes. Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$? Current best: Erd\H{o}s 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 $1\leq k\leq l$. Prize: no. OEIS: A064152. Tags: number theory, primes.