关于Pell方程解生成的子群
On the subgroup generated by solutions of Pell's equation
摘要：
丢番图方程$a^2+mb^2=c^2$的解的等价类形成无限生成的阿贝尔群$G_m$，其中$m$是固定的无平方的正整数。Pell方程$x^2-my^2=1$的解生成$G_m$的子群$P_m$。我们证明了对于所有的$m>1$，$P_m$和$G_m/P_m$具有无穷秩.我们也给出了几个例子$m$的$G_m/P_m$有非平凡挠。
英文摘要：
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.
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