KöMaL 2022 A815

Let $q$ be a monic polynomial with integer coefficients. Prove that there exists a constant $C$ depending only on polynomial $q$ such that for an arbitrary prime number $p$ and an arbitrary positive integer $N\le p$ the congruence $n! \equiv q(n) \pmod{p}$ has at most $CN^{2/3}$ solutions among any $N$ consecutive integers.