Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities

TL;DR

This paper derives formulas and identities for the sum of quadratic residues modulo primes of the form 4n-1, connecting them with roots of quadratic equations and class numbers of imaginary quadratic fields.

Contribution

It introduces new formulas and identities linking quadratic residues, roots of quadratic equations, and class numbers of imaginary quadratic fields for primes of the form 4n-1.

Findings

Derived formulas for the sum of quadratic residues modulo primes p=4n-1

Connected quadratic residues with roots of quadratic equations

Established identities involving class numbers of imaginary quadratic fields

Abstract

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for the sum of the quadratic residues are obtained. We finalize the paper by obtaining several identities involving $h(-p)$ the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p}).$