## Abstract

Let *p* be a prime and let ${p}_{1},\mathrm{\dots},{p}_{r}$ be distinct prime divisors of
$p-1$. We prove that the smallest positive integer *n* which is a
simultaneous ${p}_{1},\mathrm{\dots},{p}_{r}$-power nonresidue modulo *p* satisfies

$n<{p}^{1/4-{c}_{r}+o(1)}\mathit{\hspace{1em}}(p\to \mathrm{\infty})$

for some positive ${c}_{r}$ satisfying ${c}_{r}={e}^{-(1+o(1))r}$ as $r\to \mathrm{\infty}$.

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