Small Prime $k$th Power Residues and Nonresidues in Arithmetic Progressions

TL;DR

This paper demonstrates the existence of small prime quadratic residues and nonresidues, and extends these results to small prime $k$th power residues and nonresidues in arithmetic progressions, unconditionally for large primes.

Contribution

It establishes the existence of small prime residues and nonresidues in arithmetic progressions unconditionally, generalizing to $k$th power residues for $k o ext{log log } p$.

Findings

Existence of small prime quadratic residues and nonresidues in arithmetic progressions.

Extension of results to small prime $k$th power residues and nonresidues.

Results hold unconditionally for large primes.

Abstract

Let $p$ be a large odd prime, let $x=\log p)(\log\log p)^{3+\varepsilon}$ and let $q\ll\log\log p$ be an integer, where $\varepsilon>0$ is a small number. This note proves the existence of small prime quadratic residues and small prime quadratic nonresidues in the arithmetic progression $a+qm\ll x$, with relatively prime $1\leq a<q$, unconditionally. The same results are generalized to small prime $k$th power residues and nonresidues, where $k\mid p-1$ and $k\ll\log\log p$.