Problem Statement
Let $A(x)$ count the number of composite $u<x$ such that $n!+1\equiv 0\pmod{u}$ for some $n$. Is it true that $A(x)\leq x^{o(1)}$?
Categories:
Number Theory
Progress
A question of Erdős raised in discussions with Hardy and Subbarao [HaSu02]. The sequence of such $u$ begins\[25,121,169,437,\ldots\]and is A256519 in the OEIS.Wilson's theorem states that $u$ is prime if and only if $(u-1)!+1\equiv 0\pmod{u}$.
This is mentioned in problem A2 of Guy's collection.
Source: erdosproblems.com/1073 | Last verified: January 19, 2026