Character Sums and Congruences with n!

Abstract

We estimate character sums with n!, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime p and obtain new information about the spacings between quadratic nonresidues modulo p. In particular, we show that there exists a positive integer $n\ll p^{1/2+ε}, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a \not \equiv 0 \pmod p can be represented as a product of 7 factorials, a \equiv n_1! ... n_7! \pmod p, where $\max \{n_i | i=1,... 7\}=O(p^{11/12+ε}), and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!, with \max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+ε})$ represent ``almost all''residue classes modulo p, and that products of 3 factorials n_1!n_2!n_3! with \max\{n_1, n_2, n_3\}=O(p^{5/6+ε})$ are uniformly distributed modulo p.