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arXiv · 2506.11636

On generalised Pythagorean triples over number fields

Pedro-José Cazorla García
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Abstract

Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. 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 $\mathbb{Q}$ 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.

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Pedro-José Cazorla García. 2025-06-13. On generalised Pythagorean triples over number fields. https://arxiv.org/abs/2506.11636

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