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

Question

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?

Answer 1

this is called a CW answer; recommend beginning with

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

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