Pairing Pythagorean Pairs
Lorenz Halbeisen, Norbert Hungerb\"uhler

TL;DR
This paper explores special classes of Pythagorean pairs and their connection to elliptic curves, establishing criteria for positive rank and classifying elliptic curves with specific torsion groups related to these pairs.
Contribution
It introduces double-pythapotent and quadratic pythapotent pairs, linking them to elliptic curves with particular torsion groups and positive rank conditions, and classifies all such curves with torsion group Z/2Z×Z/8Z.
Findings
Positive rank of associated elliptic curves characterizes pythagorean pairs as double- or quadratic-pythapotent.
Every elliptic curve with torsion Z/2Z×Z/8Z is isomorphic to one derived from a pythagorean pair.
Existence of infinitely many pythagorean pairs related to double- and quadratic-pythapotent pairs.
Abstract
A pair of positive integers is a pythagorean pair if (i.e., is a square). A pythagorean pair is called a double-pythapotent pair if there is another pythagorean pair such that is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair which is not a multiple of , such that is a pythagorean pair. To each pythagorean pair we assign an elliptic curve with torsion group , such that has positive rank if and only if is a double-pythapotent pair. Similarly, to each pythagorean pair we assign an elliptic curve with torsion group , such that has positive rank if and…
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