ternary analogues of the Pell equations

Question

We know well about the Pell equations: $x^2 -ny^2=1$ and some variants of them. Criterions about the existence of nontrivial solutions of homogeneous equations $ax^2+by^2+cz^2=0$ are also well-known.

Then, how about the 3-v analogues of the Pell equations? I mean, the diophantine equations of this type: $x^2+ay^2+bz^2=1$, where $a, b $ integers. Is there an extensive survey article on them?

Comment: maybe it is trivial... in that case, what would it be on $x^2+ay^2+bz^2=n^2$? I guess these are quite nontrivial, according to some brute-force computations.

Answer 1

It turns out that Cassels does this material, pages 301-309. There is quite a big difference based on whether $x^2 - A y^2 - B z^2$ is isotropic or not, meaning there is an integer solution to $x^2 - A y^2 - B z^2=0$ with $x,y,z$ not all equal to zero. When the form is isotropic, pages 301-303, especially the proof of Lemma 5.4 and discussion on page 303.

Anisotropic is harder and the sign of the target number matters; compare Theorem 6.2 on page 305 to Theorem 6.3 on page 306.

Alright, went through the easiest example, I can see where his notation is a little different from what I expected, but he is consistent, that is what matters. In solving $x_1 x_3 - x_2^2 = 1,$ we have a single orbit, that being his $c = (1,0,1)$ from the paragraph between 5.19 and 5.20 on page 303. The result for $$ x^2 - y^2 - z^2 = 1 $$ is, with $$ \alpha \delta - \beta \gamma = 1 $$ and $$ \alpha + \beta + \gamma + \delta \equiv 0 \pmod 2, $$ $$ \left( \frac{\alpha^2 + \beta^2 + \gamma^2 + \delta^2}{2}, \; \; \frac{\alpha^2 - \beta^2 + \gamma^2 - \delta^2}{2}, \; \; \alpha \beta + \gamma \delta \right) $$

? p =   ( a^2 + b^2 + c^2 + d^2 )^2 - ( a^2 - b^2 + c^2 - d^2 )^2 - (2 * a * b + 2 * c * d   )^2
%7 = 4*d^2*a^2 - 8*d*c*b*a + 4*c^2*b^2
? q = 4 *  ( a * d - b * c)^2
%8 = 4*d^2*a^2 - 8*d*c*b*a + 4*c^2*b^2
? p - q
%9 = 0