Distribution of factorials modulo p

Abstract

We prove that the sequence $n!(\mathrm{mod}p)$ occupies at least $\sqrt{\frac{3}{2}N}$ residue classes in the short interval $H\le n\le H+N$ and $N\gg p^{\frac{1}{4}}$ improving previously known trivial bound $\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!(\mathrm{mod}p)$ for $p\le x.$

Cited by in corpus (1)

• A note on $n!$ modulo $p$