On generalised Pythagorean triples over number fields

2025/06/13 by Pedro‐José Cazorla García, García, Pedro-José Cazorla
Mathematics · #History and Theory of Mathematics #Mathematics and Applications #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2506.11636

Abstract

Generalised Pythagorean triples are integer tuples (x,y,z) satisfying the equation E_a,b,c: ax²+by²+cz²=0. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular, we can now determine whether E_a,b,c has solutions and find them in a computationally effective manner. In this paper, we consider an extension of generalised Pythagorean triples to number fields K. In particular, we survey and extend the existing results over ℚ for determining if E_a,b,c has solutions over number fields and if so, to find and parameterise them, as well as to find a minimal solution. Throughout the text, we incorporate numerous examples to make our results accessible to all researchers interested in the topic of generalised Pythagorean triples.

Citations

• On the magnitude of the gaussian integer solutions of the Legendre equation

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