Consider the diophantine equation $$ x^n+y^n+z^n=k\cdot xyz, $$ where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and $n=2$ the equation is unsolvable. This fact remains true for $n=3$, this time the proof is more elaborated and can be found in the paper Third-degree diophantine equations by M. Z. Garaev, see the link http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.8786 Hence, arise the natural question:
Q. This equation is solvable for $n\geq 4$ and $k=4$? The same question for $k=7$ and $k=8$.
Thanks in advance.