Quantizing Pythagorean triples

TL;DR

This paper introduces a novel $q$-deformation of the Pythagorean equation, creating polynomial analogues of primitive and standard triples using $q$-deformed rational numbers and modular group actions.

Contribution

It develops a new polynomial $q$-analogue of Pythagorean triples, expanding the algebraic framework for these classical geometric objects.

Findings

Constructed $q$-analogues for primitive Pythagorean triples

Extended the construction to standard Pythagorean triples

Utilized $q$-deformed rational numbers and modular group methods

Abstract

We introduce a $q$-deformation of the Pythagoras equation $a^2 + b^2 = c^2$, which is a polynomial version of it different from the standard one. We construct a polynomial analogue, or ``$q$-analogue'', of every primitive Pythagorean triple. We also construct such analogue for a larger class of Pythagorean triples called standard. Our approach is based on the notion of $q$-deformed rational numbers and the modular group $\mathrm{PSL}(2,\mathbb{Z})$.