Class Number of the Imaginary Quadratic Field and Quadratic Residues Identities

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 $\mathbb{Q}(\sqrt{-p}).$