Quantizing Pythagorean triples

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 $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z})$.