Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

TL;DR

This paper explores the algebraic structure of Pythagorean triples using complex numbers and abelian groups, providing a new effective method for enumerating triples with a given hypotenuse, accessible to a broad mathematical audience.

Contribution

It offers a novel structural description of Pythagorean triples as points on the unit circle forming an abelian group, leading to an effective enumeration method.

Findings

Structural description of Pythagorean triples as an abelian group

New effective method for generating triples with a given hypotenuse

Accessible exposition suitable for undergraduate students

Abstract

It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately, an enumeration of the normalized pythagorean triples with a given hypotenuse, and also to an effective method for producing all such triples. This effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students, and therefore it contains plenty of background material, some history and several examples and exercises.