Does the equation $x^3+y^3+2z^3=0$ have any non-trivial pairwise co-prime integer solutions?

Question

I know for any pairwise co-prime integers $x,y,z$ that $$x^3+y^3+z^3\neq 0$$ $$x^3+y^3+3z^3\neq 0$$ Do we also have $$x^3+y^3+2z^3\neq 0?$$ Any Hints?

Answer 1

There are no non-trivial solutions. A proof that there are no non-trivial solutions of the closely related equation:

$$x^3+y^3=2z^3$$

is given in Sierpinski's Elementary Theory of Numbers, Chapter 2 page 79, which may be accessed here (which I found via here). Then it just needs to be noted that a solution of either one of these equations would provide a solution of the other by change of sign of $z$.