Open AccessDistribution of factorials modulo p
Author(s) -
Oleksiy Klurman,
Marc Munsch
Publication year - 2017
Publication title -
journal de théorie des nombres de bordeaux
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.663
H-Index - 26
eISSN - 2118-8572
pISSN - 1246-7405
DOI - 10.5802/jtnb.974
Subject(s) - mathematics , modulo , combinatorics , number theory , distribution (mathematics) , arithmetic , mathematical analysis
We prove that the sequence $n!\,(\bmod\,p)$ occupies at least $\sqrt{\frac{3}{2}N}$ residue classes in the short interval $H\le n \le H+N$ and $N\gg p^{\frac{1}{4}}$ improving previously known trivial bound $\sqrt{N}.$ In the other direction, we estimate the average number of residue classes missed by the sequence $n!\,(\bmod\,p)$ for $p\le x.$
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