# Quadratic diophantine equations and geometry of numbers

Let (for concreteness) $$a = 2$$, $$b = \sqrt{5}$$ and $$\varphi = (\sqrt{5}+1)/2$$. I am interested in solutions $$(w,x,y,z) \in \mathbb{Z}[\varphi]^4$$ of the system

$$w^2 – ax^2 -by^2 + abz^2 = 1$$ $$\lvert w^2 + ax^2 +by^2 + abz^2 \rvert \ll \infty$$ $$\lvert \bar{w}^2 + a\bar{x}^2 -b\bar{y}^2 – ab\bar{z}^2 \rvert \le C$$ for some constant $$C$$. Here $$\overline{\alpha + \beta\sqrt{5}} = \alpha – \beta\sqrt{5}$$ and “$$\ll \infty$$” means that ideally I would like to enumerate solutions in increasing order of this value. (Restriction of scalars turns this problem into a system of two quadratic equations and two inequalities in eight variables in $$\mathbb{Z}$$; if someone wants to see it, I can write it out including potential mistakes).

1. What is the best (or even any practical) way to produce these?

I am aware that there is a lot of classical mathematics associated to this question but I don’t quite manage to put it together. Perhaps a subquestion is:

1. Can one enumerate the squares $$s$$ in $$\mathbb{Z}[\varphi]$$ with $$\lvert \bar{s} \rvert \le C$$ in increasing order of $$\lvert s \rvert$$?

Context: Let $$k = \mathbb{Q}(\varphi)$$ and let $$A$$ be the quaternion algebra $$(\frac{a,b}{k})$$ with norm $$\nu$$. With the above values the algebra $$A$$ is a skew field but tensoring with $$\mathbb{R}$$ in the two possible ways (taking $$\sqrt{5}$$ to $$\pm\sqrt{5}$$) gives an isomorphism with $$M_2(\mathbb{R})$$ which we equip with the map $$\left\lVert\left(\begin{array}{cc}\alpha&\beta\\gamma&\delta\end{array}\right)\right\rVert = \frac{1}{2}\left(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\right).$$ The above system then asks for solutions $$\lambda$$ in the maximal order of $$A$$ for $$\nu(\lambda) = 1$$, $$\lVert\lambda\rVert_{\sqrt{5} \mapsto \sqrt{5}} \ll \infty$$ and $$\lVert \lambda \rVert_{\sqrt{5} \mapsto -\sqrt{5}} \le C$$.