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 ?

Does this answer your question? For which values of integer $k$, does the equation $x^2+y^2+z^2=kxyz$ have positive integer solutions $(x, y, z)$ - found using an Approach0 search. There are also several AoPS threads, e.g., x^2 + y + z^2 = nxyz in positive integers, ... — John Omielan, Apr 25 '23 at 01:28
(cont.) Solution by VJ, Find k if x^2+y^2+z^2=kxyz don't have any integer solution, etc. — John Omielan, Apr 25 '23 at 01:30
see http://zakuski.math.utsa.edu/~jagy/Hurwitz_A_1907.pdf which proves it. Also https://en.wikipedia.org/wiki/Markov_number — Will Jagy, Apr 25 '23 at 02:28

score 0 · Answer 1 · answered Apr 25 '23 at 02:50

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 $n \leq 10.$

David · Answer 2 · 2023-04-25T21:30:08.740

-1

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:

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

edited Apr 25 '23 at 21:30

answered Apr 25 '23 at 14:34

David

71

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

2 Answers2