A note on $n!$ modulo $p$

TL;DR

This paper establishes a lower bound on the number of distinct factorial residues modulo a prime within a certain range and uses this to refine representations of elements modulo p as products of factorials.

Contribution

It provides a new lower bound on factorial residues in intervals and improves the known bounds for representing elements modulo p as products of factorials.

Findings

Lower bound on factorial residues in specified intervals.

Any nonzero element modulo p can be expressed as a product of seven factorials with smaller bounds.

Refinement of previous bounds for factorial representations modulo p.

Abstract

Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to show that any $\lambda\not\equiv 0\pmod p$ can be represented in the form $\lambda \equiv n_1!...n_7!\pmod p$, where $n_i=o(p^{11/12})$. This slightly refines the previously known range for $n_i$.