Exponential Sums and Congruences with Factorials

Abstract

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials $n!m!$ with $\max\{n,m\}<p^{1/2+ε}$ are uniformly distributed modulo $p$, and that any residue class modulo $p$ is representable in the form $m!n!+n_1! + ... +n_{49}!$ with $\max \{m,n, n_1, >..., n_{49}\} < p^{8775/8794+ ε}$.