The group of quasi-Pythagorean primitive triples (Q1303896)

Let \(m>1\) be a square-free integer. The set of primitive solutions of the diophantine equation \(x^2+my^2=z^2\) has a natural structure of an abelian group induced by the multiplicative group \({\mathbb Q}(\sqrt{-m})^*\). Denote this group by \(G_m\), and call it the group of quasi-Pythagorean primitive triples. Generalizing a result proposed in \textit{E. J. Eckert} [Math. Mag. 57, 22-27 (1984; Zbl 0534.10010)], the author shows that \(G_m\) is torsion-free, except for \(m=3\) (the torsion corresponds to the roots of unity in \({\mathbb Q}(\sqrt{-m})^*\) modulo \(\{\pm 1\}\)). For a finite number of cases, namely when each genus of the quadratic forms with discriminant \(-4m\) consists of a single class, the author also shows that \(G_m\) is a free abelian group of infinite rank.