On the smallest simultaneous power nonresidue modulo a prime

Abstract

AbstractLet p be a prime and let ${p_{1},\ldots,p_{r}}$ be distinct prime divisors of ${p-1}$. We prove that the smallest positive integer n which is a simultaneous ${p_{1},\ldots,p_{r}}$-power nonresidue modulo p satisfies$n<p^{1/4-c_{r}+o(1)}\quad(p\to\infty)$for some positive ${c_{r}}$ satisfying ${c_{r}=e^{-(1+o(1))r}}$ as ${r\to\infty}$.