The group of primitive almost Pythagorean triples (Q365945)

By sending a Pythagorean triple \((x,y,z)\), i.e., an integral solution of the Pythagorean equation \(x^2 + y^2 = z^2\), to the rational point \((\frac xz, \frac yz)\) one gets a bijection between primitive Pythagorean triples and rational points on the unit circle \(X^2 + Y^2 = 1\). Since the rational points on the unit circle carry a natural group structure (by identifying the pair \((X,Y)\) with the complex number \(X + Yi\)), primitive Pythagorean triples can be given a group structure by transport of structure. It is a classical result (see e.g. \textit{L. Tan} [Math. Mag. 69, No. 3, 163--171 (1996; Zbl 1044.11584)]) that the group of Pythagorean triples is not finitely generated.NEWLINENEWLINEIn this article, the authors consider triples of integers satisfying the equation \(X^2 + qY^2 = Z^2\) for positive integers \(q\). They show that the group of such triples is not finitely generated and determine their precise structure by giving their generators for \(q \in \{2, 3, 5, 6\}\).