Type: Article

A note on n! modulo p

Garaev, M. Z; Hernández, J.;

Abstract:

Let p be a prime, ? > 0 and 0 < L + 1 < L + N < p. We prove that if p1/2+?< N < p1-?, then #{n! (mod, p); L + 1 ? n ? L + N} > c (N log N)1/2, c = c(?) > 0. We use this bound to show that any ??0(modp) can be represented in the form ? ? n1!?n7!(modp), where ni= o(p11/12). This refines the previously known range for n