A note on n! modulo p

Abstract

Let p be a prime, $\varepsilon >0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon }< N <p^{1-\varepsilon }$, then

$$\begin{aligned} \#\{n!\,\,({\mathrm{mod}} \,p);\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon )>0. \end{aligned}$$

We use this bound to show that any $\lambda \not \equiv 0\ ({\mathrm{mod}}\, p)$ can be represented in the form $\lambda \equiv n_1!\cdots n_7!\ ({\mathrm{mod}}\, p)$, where $n_i=o(p^{11/12})$. This refines the previously known range for $n_i$.