On generalised Pythagorean triples over number fields

TL;DR

This paper extends the study of generalized Pythagorean triples from rational numbers to arbitrary number fields, providing methods to determine solutions, find minimal solutions, and parameterize them with numerous examples.

Contribution

It generalizes existing results over \,\mathbb{Q}\, to number fields, offering new techniques for solution determination and parameterization.

Findings

Extended solution criteria to number fields

Developed methods to find and parameterize solutions

Provided numerous illustrative examples

Abstract

Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular, we can now determine whether $E_{a,b,c}$ has solutions and find them in a computationally effective manner. In this paper, we consider an extension of generalised Pythagorean triples to number fields $K$. In particular, we survey and extend the existing results over $\mathbb{Q}$ for determining if $E_{a,b,c}$ has solutions over number fields and if so, to find and parameterise them, as well as to find a minimal solution. Throughout the text, we incorporate numerous examples to make our results accessible to all researchers interested in the topic of generalised Pythagorean triples.