On the sequence $n! \bmod p$

Alexandr Grebennikov; Arsenii Sagdeev; Aliaksei Semchankau; Aliaksei Vasilevskii

doi:10.4171/rmi/1422

JournalsrmiVol. 40, No. 2pp. 637–648

On the sequence $n! mod p$

Alexandr Grebennikov
Saint-Petersburg State University, Russia; IMPA, Rio de Janeiro, Brazil
Arsenii Sagdeev
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Aliaksei Semchankau
Moscow State University; Saint-Petersburg State University, Russia
Aliaksei Vasilevskii
Carnegie Mellon University, Pittsburgh, USA

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Abstract

We prove that the sequence $1!, 2!, 3!, \dots$ produces at least $(2 + o (1)) p$ distinct residues modulo prime $p$ . Moreover, the factorials within an interval $I \subseteq {0, 1, \dots, p - 1}$ of length $N > p^{7/8 + ε}$ produce at least $(1 + o (1)) p$ 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 $n_{i} = O (p^{6/7 + ε})$ for all $i = 1, \dots, 7$ , which provides a polynomial improvement upon the preceding results.

Cite this article

Alexandr Grebennikov, Arsenii Sagdeev, Aliaksei Semchankau, Aliaksei Vasilevskii, On the sequence $n! mod p$ . Rev. Mat. Iberoam. 40 (2024), no. 2, pp. 637–648

DOI 10.4171/RMI/1422