Distribution of factorials modulo p

TL;DR

This paper proves new lower bounds on the number of distinct residue classes taken by factorials modulo p within short intervals, and estimates the average number of residue classes missed by factorials for primes up to x.

Contribution

It improves the known bounds on the distribution of factorials modulo p in short intervals and provides average estimates for missed residue classes.

Findings

Factorials occupy at least   da in short intervals for N a p^{1/4}

Established lower bounds on the number of distinct residue classes of n! mod p

Estimated the average number of residue classes missed by factorials for primes p a x.

Abstract

We prove that the sequence $n!\,(\bmod\,p)$ 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 $n!\,(\bmod\,p)$ for $p\le x.$