On power residues modulo a prime

Ke Gong; Chaohua Jia

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arXiv · 1908.10536

On power residues modulo a prime

Ke Gong·Chaohua Jia

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Abstract

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ 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 $x^n\pmod p$ 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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Ke Gong, Chaohua Jia. 2019-08-28. On power residues modulo a prime. https://arxiv.org/abs/1908.10536

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