On power residues modulo a prime

Abstract

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|p-1$ and $n>p^{\epsilon }$, where $\epsilon$ is a sufficiently small constant. Let $k(p,n)$ denote the least positive integer $k$ such that for $x=-k,\dots ,-1,1,2,\dots ,k$, the numbers $x^n(\mathrm{mod}p)$ yield all the non-zero $n$-th power residues modulo $p$. In this paper, we shall prove $$k(p,n)=O(p^{1-\delta }),$$ which improves a result of S. Chowla and H. London in the case of large $n$.

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