A computable formula for the class number of the imaginary quadratic field $Q(\sqrt{-p}), \ p=4n-1$

TL;DR

This paper derives a computable formula for the class number of imaginary quadratic fields with primes of the form 4n-1 by analyzing quadratic residues, providing new insights into their structure and sums.

Contribution

It introduces a novel formula for the class number of Q(√-p) based on counting quadratic residues for primes p=4n-1.

Findings

Derived a formula for class number h of Q(√-p)

Established methods to count quadratic residues for primes p=4n-1

Provided formulas for sums of quadratic residues

Abstract

A formula for the class number $h$ of the imaginary quadratic field $Q(\sqrt{-p}$ is obtained by counting on a specific way the quadratic residues of a prime number of the form $p=4n-1.$ Formulas for the sum of the quadratic residues are also found.