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Article Dans Une Revue Hardy-Ramanujan Journal Année : 2023

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

Résumé

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.
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Dates et versions

hal-03914218 , version 1 (28-12-2022)

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José Manuel Rodriguez Caballero. A q-analog of Jacobi's two squares formula and its applications. Hardy-Ramanujan Journal, 2023, Volume 45 - 2022, pp.84 -- 87. ⟨10.46298/hrj.2023.10908⟩. ⟨hal-03914218⟩
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