CHARACTER SUMS AND CONGRUENCES WITH n!

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 ≪ p1/2+ε, such that n! is a primitive root modulo p. We also show that every nonzero congruence class a 6≡ 0 (mod p) can be represented as a product of 7 factorials, a ≡ n1! . . . n7! (mod p), where max{ni | 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 n1!n2!n3!n4!, with max{n1, n2, n3, n4} = O(p 6/7+ε) represent “almost all” residue classes modulo p, and that products of 3 factorials n1!n2!n3! with max{n1, n2, n3} = O(p 5/6+ε) are uniformly distributed modulo p.