Quadratic Residues and Non-residues in Arithmetic Progression

TL;DR

This paper investigates the distribution of quadratic residues and non-residues within arithmetic progressions, providing asymptotic results and exploring related generalizations and open problems.

Contribution

It determines the asymptotic behavior of quadratic residue sets in arithmetic progressions and introduces new generalizations and related problems.

Findings

Asymptotic formulas for c(p) as p approaches infinity

Extensions to various constructions of arithmetic progressions

Discussion of related open problems

Abstract

Let S be an infinite set of non-empty, finite subsets of the nonnegative integers. If p is an odd prime, let c(p) denote the cardinality of the set {T {\in} S : T {\subseteq} {1,...,p-1} and T is a set of quadratic residues (respectively, non-residues) of p}. When S is constructed in various ways from the set of all arithmetic progressions of nonnegative integers, we determine the sharp asymptotic behavior of c(p) as p {\to} +{\infty}. Generalizations and variations of this are also established, and some problems connected with these results that are worthy of further study are discussed.