On the subgroup generated by solutions of Pell's equation

Abstract

Equivalence classes of solutions of the Diophantine equation $a^2+m{b}^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-m{y}^2=1$ 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.

Corrected version, where we used a different approach based on the Frobenius density theorem to prove the infinitude of the rank of the factor group Gm/Pm for all square-free integer m>1. 14 pages