On generalised Pythagorean triples over number fields

Abstract

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Generalised Pythagorean triples are integer tuples (x, y, z) satisfying the equation Ea,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 Ea,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 Ea,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.