There exist no Fermat numbers which are cubes

Question

Prove that there exist no Fermat numbers which are cubes.

How does one go about proving this? I think the problem is from an olympiad. I've succeeded in proving that no Fermat numbers are squares, but I'm struggling for cubes. An idea I had was to prove that there exists no integer $x$ such that

$$2^{2^n} + 1 = x^3,$$

but I got stuck there. I also saw that there exists a proof for squares on ProofWiki but not for cubes.

Any tips?

Answer 1

Hint: there is a small integer $m$ such that the cubes modulo $m$ are different from the Fermat numbers modulo $m$.