The Gauss-Dirichlet Orbit Number

TL;DR

This paper completes the computation of the number of equivalence classes of representations of integers by binary quadratic forms for specific discriminants, building on Dirichlet's earlier work.

Contribution

It extends Dirichlet's previous results by providing a complete calculation of the orbit number for certain quadratic forms.

Findings

Completed the computation of the orbit number for specific discriminants

Extended Dirichlet's earlier partial results

Provided explicit formulas for the number of equivalence classes

Abstract

Dirichlet computed in some particular cases the number of equivalence classes of representations of a nonzero integer by a representative system for the integral binary quadratic forms of a given discriminant. We complete this computation.