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Find $k \in \mathbb{N}$ such that $x^3+y^3+z^3=kx^2y^2z^2$ have positive integer roots

I know a similar problem $x^3 + y^3 + z^3 = nxyz$ but I still can't solve my problem

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@D3e0X4 - If you like answers, you should accept, or at very least upvote them. If an answer isn't what you were looking for, you can use comments to clarify ehat you want to other users. – nbubis Feb 20 '13 at 15:33

Do you want all $k$, or will you settle for a couple of them?

$k=1$ works with $\{{x,y,z\}}=\{{1,2,3\}}$.

$k=3$ works for $x=y=z=1$.

Getting all solutions, that may be asking for too much. These higher degree Diophantine equations can be tough.

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