Products of quadratic residues and related identities

TL;DR

This paper investigates products of quadratic residues modulo odd primes, establishing identities involving Legendre symbols, class number formulas, and Gauss sums, thereby advancing understanding of quadratic residue properties and related number theoretic identities.

Contribution

It introduces new identities for products of quadratic residues modulo primes, connecting them with class number formulas, quartic Gauss sums, and Dirichlet L-series evaluations.

Findings

Proves a specific product identity for primes congruent to 5 mod 8.

Links quadratic residue products to class number and Gauss sums.

Provides explicit congruences involving Legendre symbols and 4th power residues.

Abstract

In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.