Open access

Date

2022-04Type

- Journal Article

ETH Bibliography

yes
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Abstract

A pair (a,b) of positive integers is a pythagorean pair if a2+b2=□ (i.e., a2+b2 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 (a2k,b2l) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γa,b with torsion group Z/2Z×Z/4Z, such that Γa,b has positive rank over Q if and only if (a,b) is a double-pythapotent pair. Similarly, to each pythagorean pair (a,b) we assign an elliptic curve Γa with torsion group Z/2Z×Z/8Z, such that Γa has positive rank over Q if and only if (a,b) is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve Γ with torsion group Z/2Z×Z/8Z is isomorphic to a curve of the form Γa, 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. Show more

Permanent link

https://doi.org/10.3929/ethz-b-000515375Publication status

publishedExternal links

Journal / series

Journal of Number TheoryVolume

Pages / Article No.

Publisher

ElsevierSubject

Pythagorean pair; Elliptic curveOrganisational unit

03874 - Hungerbühler, Norbert / Hungerbühler, Norbert
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ETH Bibliography

yes
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