Pairing powers of pythagorean pairs
Year: 2024
Type: article
Abstract: A pair ( ๐ , ๐ ) of positive integers is a pythagorean pair if ๐ 2 + ๐ 2 is a square. A pythagorean pair ( ๐ , ๐ ) is called a pythapotent pair of degree โ if there is another pythagorean pair ( ๐ , ๐ ) , which is not a multiple of ( ๐ , ๐ ) , such that ( ๐ โ โข ๐ , ๐ โ โข ๐ ) is a pythagorean pair. To each pythagorean pair ( ๐ , ๐ ) we assign an elliptic curve ๐ค ๐ โ , ๐ โ for โ โฅ 3 with torsion group isomorphic to โค / 2 โข โค ร โค / 4 โข โค such that ๐ค ๐ โ , ๐ โ has positive rank over โ if and only if ( ๐ , ๐ ) is a pythapotent pair of degree โ . As a side result, we get that if ( ๐ , ๐ ) is a pythapotent pair of degree โ , then there exist infinitely many pythagorean pairs ( ๐ , ๐ ) , not multiples of each other, such that ( ๐ โ โข ๐ , ๐ โ โข ๐ ) 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.
Source: Indagationes Mathematicae
Institution: ETH Zurich
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $1,940
OpenAlex ID: W4403272090