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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 « p 1/2+e such that n! is a primitive root modulo p. We also show that every nonzero congruence class a? 0 (mod p) can be represented as a product of 7 factorials, a ≡ n 1 !...n 7 ! (mod p), where max{n i } i = 1,...,7} = O(p 11/12+e ), 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+e ) 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+e ) are uniformly distributed modulo p.

Character sums and congruences with 𝑛!