A note on $n!$ modulo $p$

Abstract

Let $p$ be a prime, $\epsilon >0$ and $0<L+1<L+N<p$. We prove that if $p^{1/2+\epsilon }<N<p^{1-\epsilon }$, then $$\#\{n!(\mathrm{mod}p);L+1\le n\le L+N\}>c(N\log N{)}^{1/2},c=c(\epsilon )>0.$$ We use this bound to show that any $\lambda \not\equiv 0(\mathrm{mod}p)$ can be represented in the form $\lambda \equiv n_1!...n_7!(\mathrm{mod}p)$, where $n_i=o(p^{11/12})$. This slightly refines the previously known range for $n_i$.

9 pages

References in corpus (1)

• Distribution of factorials modulo p