Sums of Quadratic residues and nonresidues

TL;DR

This paper explores the sums of quadratic residues and nonresidues modulo primes, extending known results from primes congruent to 1 mod 4 to those congruent to 3 mod 4, revealing new relationships.

Contribution

It provides new analogous results for primes congruent to 3 mod 4, expanding the understanding of quadratic residues and nonresidues sums.

Findings

Established sum relationships for quadratic residues when p ≡ 3 mod 4

Extended known results from p ≡ 1 mod 4 case to p ≡ 3 mod 4

Contributed to number theory by analyzing quadratic residues sums

Abstract

It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.