Generating B\'ezout trees for Pythagorean pairs

TL;DR

This paper introduces a method to generate Bézout trees for Pythagorean pairs, enabling efficient computation and analysis of coprime integer pairs with applications in graphics and number theory.

Contribution

The paper constructs Bézout coefficient trees for Pythagorean pairs and demonstrates their use in comparing gcd computations and reducing hyperbolic wallpaper design processing time.

Findings

Bézout trees accurately represent coprime pairs.

Comparison shows Bézout coefficients align with Matlab's gcd results.

Application reduces computation time in graphic design tasks.

Abstract

Relatively prime pairs of integers can be represented as nodes in three way branching trees. We construct trees of B\'ezout coefficients which correspond to the relatively prime pairs in the aforementioned trees. As one application, we compare the B\'ezout coefficients in these trees to those returned by the gcd function in Matlab. As another application, we use these trees to decrease the computation time required to create computer generated hyperbolic wallpaper designs.