Q:

On the notion of primary representation of a natural number by a quadratic form

This "discussion" has to do with some of the material we can find in pages 183-186 of the translation into English of the first part of E. Landau's Vorlesungen über Zahlentheorie (published by the Chelsea Publishing Co. in 1966).

Let $k \in \mathbb{N}$ and $F=\{a,b,c\}$, where $a>0$, be an integral binary quadratic form of discriminant $d>0$ such that $F(x,y)=k$ for some $x, y \in \mathbb{Z}$. We say that the previous representation of $k$ by $F$ is primary if

$$2ax+(b-\sqrt{d})y>0, \quad 1 \leq \frac{2ax+(b+\sqrt{d})y}{2ax+(b-\sqrt{d})y} <\varepsilon^{2}$$
where $\varepsilon$ is the fundamental solution of the equation $t^{2}-du^{2}=4$.
I believe that Landau defines a primary representation of a natural number $k$ by a form $F$ in this somewhat peculiar way because it will allow him to single out, for every $\ell$ such that
$$\ell^{2} \equiv d \pmod{4k}, \quad 0\leq \ell < 2k,$$
exactly one (canonical?) primary representation of $k$.
What can you tell me about this appreciation of mine? Is it dead wrong? How do you usually motivate this definition when you teach out of this famous book?

In pages 184-185, Landau proves the following:

Let us set 
$$w= \left\{ \begin{array}{cc} 
1 & \text{ if } d > \,\,\,\, 0 \\
2 & \text{ if } d < -4\\
4 & \text{ if } d=-4\\
6 & \text{ if } d=-3
\end{array}\right.$$
Then, for every $\ell$ such that
\begin{eqnarray}\ell^{2} \equiv d \pmod{4k},  \quad 0\leq \ell <2k,\end{eqnarray}
there are exactly $w$ proper primary representations of $k$ to which $\ell$ is associated (at this point of his exposition, Landau has already taught how to associate  to any proper representation of $k$ by a form a unique solution of $\ell^{2} \equiv d \pmod{4k}, 0\leq \ell <2k$, cf. Thm. 201 on page 180). 
I consider that, for negative $d$'s, this assertion can be proved more succinctly as follows.
Let us fix and $\ell$ such that $\ell^{2} \equiv d \pmod{4k}, 0 \leq \ell < 2k$. Then, if $m:= \frac{\ell^{2}-d}{4k}$ and $G :=\{k,l,m\}$, it follows that $G \sim F_{i}$ for exactly one form $F_{i}$ in a complete set of representatives of the equivalence classes of primitive and positive-definite forms of discriminant $d$. Then, it is not difficult to convince oneself that there is a bijective correspondence between the proper primary (proper primary = proper, in the case of negative discriminants) representations of $k$ to which $\ell$ is associated and the matrices $g\in \mathrm{SL}_{2}(\mathbb{Z})$ such that $gF_{i} = G$. Hence, since any such $g$ can be written in the form $g_{0}^{-1}A$ for a fixed $g_{0} \in \mathrm{SL}_{2}(\mathbb{Z})$ such that $g_{0}G = F_{i}$ and any automorph $A$ of $F_{i}$, it does follow that the number of proper representations of $k$ to which $\ell$ is associated is equal to  the number of different automorphs of $F_{i}$ (which is exactly $w$).
I suppose that Landau's proof turns out to be a wee bit long-winded because he is trying to handle simultaneously the cases $d<0$ and $d>0$, right? 
Please, let me thank you in advance for your attentive replies, comments, bibliographical suggestions, etc.

A:

I'm going to take a more algebraic number theoretic approach to this question. The solutions to $x^2-dy^2=k$ are the integers of norm $k$ in the number field $\mathbb{Z}[\sqrt(d)]$ if $d$ is a fundamental discriminant. Such integers are acted on by the unit group. To have a unique solution one must mod out by this action, but Landau singles out a unique representative instead. Those strange numbers $w$ are just the number of torsion units in various quadratic fields.

