Pairing Powers of Pythagorean Pairs

TL;DR

This paper explores the concept of pythapotent pairs of Pythagorean pairs, establishing a connection with elliptic curves and demonstrating the existence of infinitely many related pairs, especially highlighting that all Pythagorean pairs are pythapotent of degree 3.

Contribution

It introduces the notion of pythapotent pairs of degree h, links them to elliptic curves with specific torsion groups, and proves that all Pythagorean pairs are pythapotent of degree 3, expanding previous studies.

Findings

Existence of elliptic curves with positive rank linked to pythapotent pairs

All Pythagorean pairs are pythapotent of degree 3

Infinite families of Pythagorean pairs related to pythapotent pairs

Abstract

A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$, such that $(a^hk, b^hl)$ is a pythagorean pair. To each pythagorean pair $(a, b)$ we assign an elliptic curve $\Gamma_{a^h ,b^h}$ for $h\ge 3$ with torsion group isomorphic to $\mathbb Z/2\mathbb Z \times \mathbb Z/4\mathbb Z$ such that $\Gamma_{a^h,b^h}$ has positive rank over $\mathbb Q$ if and only if $(a,b)$ is a pythapotent pair of degree $h$. As a side result, we get that if $(a, b)$ is a pythapotent pair of degree $h$, then there exist infinitely many pythagorean pairs $(k,l)$, not multiples of each other, such that $(a^hk,b^hl)$ is a pythagorean pair. In particular, we show that any pythagorean pair is always a pythapotent pair of degree 3. In a previous work, pythapotent pairs of degrees 1 and 2 have been studied.