On generalised Pythagorean triples over number fields
Abstract
. 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.