A $q$-analog of Jacobi's two squares formula and its applications

TL;DR

This paper introduces a $q$-analog of Jacobi's two squares formula, generalizing classical results, and explores its applications in characterizing hypotenuse lengths of primitive Pythagorean triangles.

Contribution

It develops a new $q$-analog of the sum of two squares representation formula and applies it to number theory problems involving Pythagorean triangles.

Findings

Generalized Jacobi's formula to $r_2(n, q)$

Characterized signs in coefficients via prime factors

Identified integers as hypotenuse lengths of primitive Pythagorean triangles

Abstract

We consider a $q$-analog $r_2(n, q)$ of the number of representations of an integer as a sum of two squares $r_2(n)$. This $q$-analog is generated by the expansion of a product that was studied by Kronecker and Jordan. We generalize Jacobi's two squares formula from $r_2(n)$ to $r_2(n, q)$. We characterize the signs in the coefficients of $r_2(n, q)$ using the prime factors of $n$. We use $r_2(n, q)$ to characterize the integers which are the length of the hypotenuse of a primitive Pythagorean triangle.