On the sequence $n! \bmod p$

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.