# Pairing Pythagorean Pairs

Lorenz Halbeisen, Norbert Hungerbühler

Lorenz Halbeisen, Norbert Hungerbühler

**20/1/21**Published in : arXiv:2101.08163

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,b} 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.