On the compatibility of B\'ezout coefficients between Pythagorean pairs under unimodular transformations

TL;DR

This paper proves a theorem and confirms a conjecture regarding the generation of Bézout trees for Pythagorean pairs, providing a simple proof and a generalization, thus advancing understanding of their algebraic structure.

Contribution

It offers a simple proof of Gullerud and Walker's theorem, confirms their conjecture, and generalizes the results on Bézout coefficients for Pythagorean pairs.

Findings

Theorem proved with a simplified approach

Conjecture confirmed as true

Generalization of previous results provided

Abstract

In a recent preprint, Gullerud and Walker [2] proved a theorem and made a conjecture about the correctness of efficiently generating B\'ezout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, confirm that their conjecture is true, and furthermore we give a generalization.