$\frac{x^3+y^3+z^3}{x+y+z}\in \mathbb N$ has infinitely many non-trivial solutions Trying to solve this I find out the following problem in which it is not necessary the condition $x^3=y^3=z^3$ in some $\mathbb F_p$:
Prove there are infinitely many pairwise coprime triples of distinct natural numbers, $(x,y,z)$, such that:
$$\frac{x^3+y^3+z^3}{x+y+z}\in \mathbb N$$

Let $x=y=z$ then:
$$
\frac {x^3+y^3+z^2}{x+y+z}=\frac {3x^3}{3x}=x^2\in \Bbb N, \forall x\in\Bbb Z-\{0\}
$$
Furthermore, let $z=0,y=1$, then:
$$
\frac {x^3+y^3+z^2}{x+y+z}=\frac {x^3+1}{x+1}=x^2-x+1\in\Bbb N, \forall x\in\Bbb N-\{1\}
$$
