Problem Statement
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 $p$?
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 $p$?
Categories:
Number Theory
Progress
Questions formulated by Erdős, Hardy, and Subbarao [HaSu02], who believed that the number of $p\leq x$ for which $f(p)=p-1$ is $o(x/\log x)$.These are mentioned in problem A2 of Guy's collection.
Source: erdosproblems.com/1072 | Last verified: January 19, 2026