The Group of Primitive Almost Pythagorean Triples

TL;DR

This paper studies the algebraic structure of groups formed by primitive solutions to the equation x^2 + qy^2 = z^2 for specific square-free positive integers q, revealing detailed group properties for q in {2,3,5,6}.

Contribution

It provides a complete analysis of the group structure of primitive almost Pythagorean triples for selected values of q, extending understanding of their algebraic properties.

Findings

The set of solutions forms an abelian group under complex multiplication.

Explicit group structures are characterized for q in {2,3,5,6}.

The analysis reveals differences in structure depending on q.

Abstract

We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and give a complete analysis when $q\in\{2,3,5,6\}$.