The Origin of the Beauregard--Suryanarayan Product on Pythagorean Triples
Shaul Zemel
TL;DR
This paper explores the algebraic structure of Pythagorean triples, linking it to hyperbolic plane automorphisms and Pell's equation, revealing new geometric and number-theoretic insights.
Contribution
It demonstrates how the Beauregard--Suryanarayan product on Pythagorean triples originates from hyperbolic automorphisms and generalizes the involution related to Pell's equation.
Findings
The product structure arises from hyperbolic automorphisms.
The involution on triples relates to Pell's equation with parameter 2.
A generalization of the involution phenomenon is proved.
Abstract
We show how the multiplicative structure on Pythagorean triples defined by Beauregard--Suryanarayan arises from automorphisms of the hyperbolic plane, and how the structure theorem of this product follow naturally from this connection. We also show how the natural involution on Pythagorean triples is related to Pell's equation with the parameter 2, and prove a generalization of this phenomenon.
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