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 maxfn; mg < p 1=2þe are uniformly distributed modulo p, and that any residue class modulo p is representable in the form m!n! þ n 1 ! þ Á Á Á þ n 47 ! with maxfm; n; n 1 ; . . . ; n 47 g < p 1350=1351þe .

Waring type congruences involving factorials modulo a prime

“…In the present paper we improve some of the results from [6] and [7]. In [6] it is shown that any residue class λ modulo p can be represented in the form…”

“…Some progress in this direction has been made in [5,6]. In particular, multiplicative character sums and double exponential sums involving factorials have been estimated.…”

Open Problems on Exponential and Character Sums

“…and single multiplicative character sums as above with a = 0 are estimated in [68] and [69], respectively. and…”

Waring problem with factorials

“…Here, we continue to study exponential sums and various congruences with factorials which have been considered in [l, 2,3,5,6,7,8] and use some results and methods of [2,3] to study the analogue of the Waring problem with factorials.…”