Type: Article

Waring type congruences involving factorials modulo a prime

M. Z. Garaev; V. C. Garcia;

Abstract:

In this paper we prove that any residue class A modulo a large prime number p can be represented in the form [GRAPHICS] for some positive integers m(1),n(1),...,m(5),n(5) of the size O(p(27/28)). This improves one of the results from [6] on represent ability of A modulo p in the form [GRAPHICS] with Max(1)<= i <= 7{m(i), n(i)} = O(p(33/34)). We also prove that any residue class modulo p can be represented in the form n(1)! +...+ n(l)! (mod p) with l = O(log(3) p). This improves the result of [7].