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For which values of integer $k$, does the equation $x^2+y^2+z^2=kxyz$ have positive integer solutions $(x, y, z)$

I immediately thought of saying that from symmetry we have that $x\le y \le z$.

Also, $y^2+z^2 \equiv 0 mod x$, $x^2+z^2\equiv 0mody$ and $x^2+y^2\equiv 0modz$.

Moreover through trial and error I worked out that the solutions for $k$ must be $k=1$ or $k=3$ but I have not managed to prove it. I attempted to use inequalities, but that didn't work out either. Could you please explain to me how to solve this question?

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  • $\begingroup$ For $k=2$ there a re no solutions, see this post. The case $k=3$ is called "Markoff's equation". Obviously then $(x,y,z)=(1,1,1)$ is a solution. $\endgroup$ Jan 9, 2021 at 14:24
  • $\begingroup$ The triple $(5,29,433)$ is a solution for $k=3$, and there are many more. There are no solutions $x\leq y\leq z\leq1000$ with $k\geq4$. Maybe you should try to prove that $k\geq4$ is impossible. $\endgroup$ Jan 9, 2021 at 14:34
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    $\begingroup$ @ChristianBlatter complete proof in zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf $\endgroup$
    – Will Jagy
    Jan 9, 2021 at 15:51
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    $\begingroup$ @DietrichBurde complete proof in zakuski.utsa.edu/~jagy/Hurwitz_A_1907.pdf I have been able to use the main idea, that of looking for fundamental solutions, for many problems on this site that amount to "Vieta Jumping." $\endgroup$
    – Will Jagy
    Jan 9, 2021 at 15:56
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    $\begingroup$ @WillJagy Very nice, thank you! Even in German! $\endgroup$ Jan 9, 2021 at 17:51

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this is called a CW answer; recommend beginning with

Equation with Vieta Jumping: $(x+y+z)^2=nxyz$.

Here are a number of posts about Vieta Jumping/ Hurwitz/Markov Grundlösung

Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.

Diophantine quartic equation in four variables

Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers? !!!!!!!!! +++++++

http://math.stackexchange.com/questions/1411049/if-a-b-are-positive-integers-and-ab-1-mid-a%C2%B2-b%C2%B2-then-prove-that-q

Proving that all terms in sequence are positive integers (Recursive)

Diophantine equation $(x+y)(x+y+1) - kxy = 0$

Find the postive integers such $xy+x+y\mid x^2+y^2-2$

Find the integer values of c

Find all solutions to the diophantine equation $(x+2)(y+2)(z+2)=(x+y+z+2)^2$

Characterize the integers $a,b$ satisfying: $ab-1|a^2+b^2$

Equation with Vieta Jumping: $(x+y+z)^2=nxyz$. Follows Hurwitz very closely!! (x+y+z)^2 = nxyz

Showing that $m^2-n^2+1$ is a square LEMMA +

Showing that $m^2-n^2+1$ is a square LEMMA -

How prove infinitely many postive integers triples $(x,y,z)$ such $(x+y+z)^2+2(x+y+z)=5(xy+yz+zx)$

Find all possible value of c

What are the solutions of the equation $3np+3n+2=n^2+p^2$, with n and p positive integers?

Prove there is no $x, y \in \mathbb Z^+ \text{ satisfying } \frac{x}{y} +\frac{y+1}{x}=4$

Integer points on a hyperbola two spiral arms

$m+n+p-1=2\sqrt{mnp}$ in positive integers

https://artofproblemsolving.com/community/c6h1726220_if_fraction_integer_then_equal_to_5 Poland 1991 training

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