A note on $n!$ modulo $p$

M. Z. Garaev, J. Hernández

Let $p$ be a prime, $\epsilon >0$ and $0<L+1<L+N<p$. We prove that if $p^{1/2+\epsilon }<N<p^{1-\epsilon }$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\lo…