Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples

Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In [5] we generalized these methods and results to Pell’s equation. We find a similar group structure and count on the number of solutions for a given z to \(x^2 + Dy^2 = z^2\) when D is 1 or 2 modulo 4 and the class group of \(\mathbb {Q}[\sqrt{-D}]\) is a free \(\ensuremath {\mathbb {Z}}_2\) module, which always happens if the class number is at most 2. In this paper we discuss the main results of [5] using some concrete examples in the case of \(D=105\) .