Keywords
., Pythagorean triples, Diophantine equations, number fields, LLL algorithm, computational number theory.
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.
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.
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