Equation in 4 variables

Question

Determine all the the solutions of the equation $x^3+y^3+z^3=wx^2y^2z^2$ in natural numbers $x,y,z,w$.

I have tried numerous approaches including divisibility and also bounding by assuming WLOG $x\geq y\geq z$, but this is leading nowhere.

Answer 1

Forget the divisibility, think again about bounding.

Indeed, we may assume WLOG that $x\geqslant y\geqslant z$. Now, the LHS is divisible by $x^2$, which $x^3$ obviously is, hence $y^3+z^3$ must be too, hence $y^3+z^3\geqslant x^2$, hence $y^3\geqslant {1\over2}x^2$, or $y^2\geqslant{1\over2^{2/3}}x^{4/3}$. I believe we can find a stricter bound, but this is quite enough already.

$$3x^3\geqslant x^3+y^3+z^3=wx^2y^2z^2\geqslant x^2y^2z^2\geqslant x^2y^2\geqslant{1\over2^{2/3}}x^{10/3}$$ $$3\cdot2^{2/3}\geqslant x^{1/3}$$ $$x\leqslant108$$ A brute force search will do the rest.