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.