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

Abstract

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. We generalize 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 $x^2+D{y}^2=z^2$ 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, as well as an example with different behavior when the class group does not have this structure.

13 pages, 1 figure