Pairing Pythagorean Pairs

Abstract

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2 = \Box$ (i.e., $a^2 + b^2$ is a square). A pythagorean pair $(a, b)$ is called a double-pythapotent pair if there is another pythagorean pair $(k,l)$ such that $(ak,bl)$ is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair $(k,l)$ which is not a multiple of $(a,b)$, such that $(a^2k,b^2l)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $\Gamma_{a,b}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/4\mathbb Z$, such that $\Gamma_{a,b}$ has positive rank if and only if $(a, b)$ is a double-pythapotent pair. Similarly, to each pythagorean pair $(a, b)$ we assign an elliptic curve $\Gamma_{a^2 ,b^2}$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$, such that $\Gamma_{a^2,b^2}$ has positive rank if and only if $(a,b)$ is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve $\Gamma$ with torsion group $\mathbb Z/2\mathbb Z\times\mathbb Z/8\mathbb Z$ is isomorphic to a curve of the form $\Gamma_{a^2 ,b^2}$ , where $(a,b)$ is a pythagorean pair. As a side-result we get that if $(a,b)$ is a double-pythapotent pair, then there are infinitely many pythagorean pairs $(k, l)$, not multiples of each other, such that $(ak, bl)$ is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs.