Let $S$ be the set of all $m\ge 1$ such that there exists a prime $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$. Does$$\lim \frac{|S\cap [1,x]|}{x}$$exist? What is it?Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1(\mathrm{mod}m)$ such that $m!+1\equiv 0(\mathrm{mod}p)$, then does$$\lim \frac{|P\cap [1,x]|}{\pi (x)}$$exist? What is it?