О наименьшем числе, являющемся одновременно степенным невычетом для некоторых степеней

I will talk about a joint paper with K.Ford and M.Z. Garaev. The main result is the following one. Let $p$ be a prime, $p_1,\dots,p_r$ be some of prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is simultaneous $p_1,\dots,p_r$-power nonresidue modulo $p$ satisfies
$$n<p^{1/4-c^{-r}},\quad n\ge n(r),$$
where $c>0$ is an absolute constant.