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, the factorials within 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! \cdots 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.