On the sequence $n! \bmod p$

TL;DR

This paper proves that factorial sequences modulo a prime produce many distinct residues and that every non-zero residue can be expressed as a product of seven factorials, improving previous bounds significantly.

Contribution

It establishes new lower bounds on the number of distinct factorial residues modulo prime and shows every non-zero residue can be represented as a product of seven factorials with polynomially bounded factors.

Findings

Factorials produce at least (\sqrt{2}+o(1))\sqrt{p} distinct residues

Intervals of factorials of length > p^{7/8 + \varepsilon} produce at least (1+o(1))\sqrt{p} distinct residues

Every non-zero residue modulo p can be expressed as a product of seven factorials with factors O(p^{6/7+\varepsilon})

Abstract

We prove, that the sequence $1!, 2!, 3!, \dots$ produces at least $(\sqrt{2} + o(1))\sqrt{p}$ distinct residues modulo prime $p$. Moreover, factorials on an interval $\mathcal{I} \subseteq \{0, 1, \dots, p - 1\}$ of length $N > p^{7/8 + \varepsilon}$ produce at least $(1 + o(1))\sqrt{p}$ distinct residues modulo $p$. As a corollary, we prove that every non-zero residue class can be expressed as a product of seven factorials $n_1! \dots n_7!$ modulo $p$, where $n_i = O(p^{6/7+\varepsilon})$ for all $i=1,\dots,7$, which provides a polynomial improvement upon the preceding results.