When do three cubics form an arithmetic progression?

Question

Are there any solutions to the diophantine equation $x^3+y^3=2z^3$ other than the trivial ones?

What about $x^4+y^4=2z^4$? I think I remember these equations in one of Euler's work, but having trouble finding it.

Any help or reference would be much appreciated.

Answer 1

No nontrivial primitive solutions to $x^n + y^n = 2 z^n$ for any $n \ge 3$. See H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s Last Theorem, J. Reine Angew. Math. 490 (1997), 81–100.