On the smallest simultaneous power nonresidue modulo a prime

TL;DR

This paper establishes an upper bound on the smallest positive integer that is a simultaneous nonresidue for multiple prime divisors of p-1, improving understanding of power nonresidues in number theory.

Contribution

It provides a new upper bound on the smallest simultaneous power nonresidue modulo a prime, incorporating the structure of prime divisors of p-1.

Findings

Bound n < p^{1/4 - c_r + o(1)} for large p

c_r ≥ e^{-(1+o(1))r} as r→∞

Advances understanding of simultaneous power nonresidues

Abstract

Let $p$ be a prime and $p_1,\ldots, p_r$ be distinct prime divisors of $p-1$. We prove that the smallest positive integer $n$ which is a simultaneous $p_1,\ldots,p_r$-power nonresidue modulo $p$ satisfies $$ n<p^{1/4 - c_r+o(1)}\quad(p\to\infty) $$ for some positive $c_r$ satisfying $c_r\ge e^{-(1+o(1))r} \; (r\to \infty).$