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Two well-studied Diophantine equations are those of Pythagorean triples andelliptic curves; for the first, we have a parametrization through rationalpoints on the unit circle, and for the second we have a structure theorem forthe group of rational solutions. Recently Yekutieli discussed a connectionbetween these two problems and described the group structure of Pythagoreantriples and the number of triples for a given hypotenuse. We generalize thesemethods and results to Pell's equation. We find a similar group structure andcount 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$\mathbb{Z}_2$ module, which always happens if the class number is at most 2.We give examples of when the results hold for a class number greater than 2, aswell as an example with different behavior when the class group does not havethis structure.

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