Integers solutions of modified Markoff equation with parameter $k\in\mathbb{N}$ : $p^2 + q^2 + r^2 = kpqr$

Question

I recently studied the Markoff equation : \begin{equation*} p^2 + q^2 + r^2 = 3pqr \end{equation*} And I think I've heard that there are solutions iff $k = 1$ or $3$ but I can't prove it. I manage to show that there are solutions of the form $(p,p,r)$ iif $k = 1$ or $3$ but that's not enough.

Any idea ?

Answer 1

Here is the page of outcomes in the 1907 article by Hurwitz. He has already shown that the collection of integer solutions to $$x_1^2 + \cdots x_n^2 = x x_1 x_2 \ldots x_n$$ arises from a single fundamental solution; those are listed for $n \leq 10.$

Answer 2

For, $k=3$

$x^2+y^2+z^2=3xyz$

Above has parametric solution:

$x=2w^2-2w+1$

$y=5w^2-4w+1$

$z=29w^4-48w^3+34w^2-12w+2$

For, $w=2$ we get:

$(x,y,z)=(5,13,194)$

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