Binary quadratic forms and the factorization method of Gauss

TL;DR

This paper explores Gauss's method of factorization using binary quadratic forms, clarifying complex computations from Gauss's original work to enhance understanding of his revolutionary approach.

Contribution

It provides a detailed analysis and computational clarification of Gauss's factorization method based on binary quadratic forms from Disquisitiones Arithmeticae.

Findings

Revealed the detailed computational steps in Gauss's method

Enhanced understanding of quadratic form composition and factorization

Clarified Gauss's approach to quadratic residues and discriminants

Abstract

In Disquisitiones Arithmeticae, Gauss studied binary quadratic forms and introduced a very general version of a composition operator that allows composing even forms of different discriminants and imprimitive forms. Section V of Disquisitiones contains truly revolutionary ideas and involves very complicated computations, sometimes left to the reader. With this theory, Gauss develops a method of factorization using quadratic residues obtained by binary quadratic forms. In this work we explore Gauss method making many of the computations necessary to better understand his work.