Pairing powers of pythagorean pairs

Abstract

A pair (a,b) of positive integers is a pythagorean pair if a2+b2 is a square. A pythagorean pair (a,b) is called a pythapotent pair of degreeh if there is another pythagorean pair (k,l), which is not a multiple of (a,b), such that (ahk,bhl) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γah,bh for h≥3 with torsion group isomorphic to Z/2Z×Z/4Z such that Γah,bh has positive rank over 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 (ahk,bhl) 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.