Class Numbers and Pell's Equation $x^2 + 105y^2 = z^2$

TL;DR

This paper explores the structure and enumeration of solutions to Pell's equation with specific parameters, extending previous work on related Diophantine equations and illustrating results with concrete examples for D=105.

Contribution

It generalizes the group structure and counting methods for solutions of Pell's equation to the case D=105, linking class group properties to solution counts.

Findings

Established a group structure for solutions when D=105.

Counted solutions for specific hypotenuse values.

Connected class number properties to solution enumeration.

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. In arXiv:2112.03663 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 $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$.