Exercise 3.7.1

Proof. Here $a=0$, and $\mathrm{\Delta }(a,b,c)=b^2$.

Note that $\mathcal{}F(0,b)=\{(0,0,c)\Gamma \mid c\in \mathbb{Z}\}$ is infinite, since $(0,0,c)\Gamma =\{(0,0,c)\}$. Therefore we can assume $\mathrm{\Delta }\ne 0$.

If $\mathrm{\Delta }$ is not a square in $\mathbb{Z}$, then $\mathcal{}F(\mathrm{\Delta },0)=\varnothing$. Assume now that $\mathrm{\Delta }={\delta }^2$ is a square of some $\delta \in \mathbb{Z},\delta >0$.

For all $(b,c)\in {\mathbb{Z}}^2$, $(0,b,c)\in \mathcal{}F(\mathrm{\Delta },0)$ if and only if $\mathrm{\Delta }(0,b,c)=b^2=\mathrm{\Delta }$, that is $b^2={\delta }^2,b=\pm \delta$, thus every $(0,\pm \delta ,c)\Gamma$ is in $\mathcal{}F(\mathrm{\Delta },0)$ (but are not all distinct).

Moreover,

Therefore every element in $\mathcal{}F(\mathrm{\Delta },0)$ is given by $(0,\pm \delta ,c)$, where $0\le c<\delta$. The elements of $\mathcal{}F(\mathrm{\Delta },0)$ are

We show that all these elements are distinct.

If $(0,\delta ,c^{\prime })\Gamma =(0,\delta ,c)\Gamma$, where $0\le c\le c^{\prime }<\delta$, then $0,\delta ,c^{\prime })=(0,\delta ,c)T^k=(0,\delta ,\mathit{b\delta }+c$, thus $c^{\prime }=\mathit{b\delta }+c$, so $\delta \mid c^{\prime }-c$, where $0\le c^{\prime }-c<\delta$, therefore $c^{\prime }=c$.

If $(0,\delta ,c^{\prime })\Gamma =(0,-\delta ,c)\Gamma$, then $0,\delta ,c^{\prime })=(0,-\delta ,c)T^k=(0,-\delta ,b-\delta +c)$, thus $\delta =-\delta$ and $\mathrm{\Delta }=0$: this contradicts the hypothesis $\mathrm{\Delta }\ne 0$.

As a conclusion, if $\mathrm{\Delta }\ne 0$, then

$\text{∙}$

If $\mathrm{\Delta }$ is not a perfect square, $\mathcal{}F(\mathrm{\Delta },0)=\varnothing$.

$\text{∙}$

If $\mathrm{\Delta }={\delta }^2,\delta >0$, then $R(\mathrm{\Delta },0)=2\delta$, and $$\begin{array}{llll}\hfill \mathcal{}F(\mathrm{\Delta },0)=& \{(0,\delta ,0)\Gamma ,(0,\delta ,1)\Gamma ,(0,\delta ,2)\Gamma ,\dots ,(0,\delta ,\delta -1)\Gamma ,& \hfill & \\ \hfill & (0,-\delta ,0)\Gamma ,(0,-\delta ,1)\Gamma ,(0,-\delta ,2)\Gamma ,\dots ,(0,-\delta ,\delta -1)\Gamma \}.& \hfill & \end{array}$$