On the subgroup generated by solutions of Pell's equation

TL;DR

This paper investigates the structure of groups generated by solutions to Pell's equation and related Diophantine equations, proving infinite rank for certain subgroups and exploring torsion properties.

Contribution

It establishes that the subgroup generated by Pell's solutions and the quotient group both have infinite rank for all square-free positive integers greater than one.

Findings

Both P_m and G_m/P_m have infinite rank for all m>1.

Examples of m where G_m/P_m has nontrivial torsion are provided.

Abstract

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.