On the smallest simultaneous power nonresidue modulo a prime

Abstract

Let $p$ be a prime and $p_1,\dots ,p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\dots ,p_r$-power nonresidue modulo $p$ satisfies $$n<p^{1/4-c_r+o(1)}(p\to \infty )$$ for some positive $c_r$ satisfying $c_r\ge e^{-(1+o(1))r}(r\to \infty ).$

15 pages