On the sequence $n!\mathrm{mod}p$

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 $N>p^{7/8+\epsilon }$ 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 $n_i=O(p^{6/7+\epsilon })$ for all $i=1,\dots ,7$, which provides a polynomial improvement upon the preceding results.

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