Hardy-Ramanujan Journal |

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.

Source: HAL:hal-03914218v1

Volume: Volume 45 - 2022

Published on: February 6, 2023

Imported on: February 6, 2023

Keywords: Jacobi's two squares formula,q-analog,primitive Pythagorean triplet,11C08, 11E25, 11N13,[MATH]Mathematics [math]

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