On power residues modulo a prime

TL;DR

This paper investigates the distribution of n-th power residues modulo a large prime p, establishing an upper bound on the minimal integer generating all residues, thus advancing previous results for large n.

Contribution

The paper proves that the minimal integer k(p,n) generating all non-zero n-th power residues modulo p is bounded by O(p^{1-δ}), improving prior bounds for large n.

Findings

k(p,n) = O(p^{1-δ}) for large n

Improves previous bounds by Chowla and London

Advances understanding of power residues distribution

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 $n$.