The group of primitive almost pythagorean triples

Abstract

We consider the triples of integer numbers that are solutions of the equation $x^2+q{y}^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and describe all generators for each $q\in \left\{2,3,5,6\right\}$. We also show that if the group has a generator with the third coordinate being a power of 2, such generator is unique up to multiplication by $\pm 1$.