Problem 1074

Erdős Problem #1074: declared status 'open'. Formalized: yes. Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$. Does$$\lim \frac{\lvert S\cap [1,x]\rvert}{x}$$exist? What is it? Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m!+1\equiv 0\pmod{p}$, then does$$\lim \frac{\lvert P\cap [1,x]\rvert}{\pi(x)}$$exist? What is it? Current best: The frequency with which the EHS numbers occur - most often in long sequences of consecutive integers - makes us believe that their asymptotic density exists and is unity. Prize: no. OEIS: A063980, A064164. Tags: number theory.