Problem 1072

Erdős Problem #1072: declared status 'open'. Formalized: yes. For any prime $p$, let $f(p)$ be the least integer such that $f(p)!+1\equiv 0\pmod{p}$. Is it true that there are infinitely many $p$ for which $f(p)=p-1$? Is it true that $f(p)/p\to 0$ for almost all $p$? Current best: Questions formulated by Erd\H{o}s, Hardy, and Subbarao [HaSu02], who believed that the number of $p\leq x$ for which $f(p)=p-1$ is $o(x/\log x)$. Prize: no. OEIS: A072937, A073944, A154554. Tags: number theory.