As a generalisation to the equation of Fermat, one can ask for rational solutions of $X^n+Y^n+Z^n=1$ (or almost equivalently integer solutions of $X^n+Y^n+Z^n=T^n$).

Contrary to the case of Fermat, the case where $n=3$ has infinitely many solutions, because the surface is rational. For $n=4$, we get a K3 surface and for $n\ge 5$ the surface should have finitely many points or the points should be contained in finitely many curves since it is of general type (Bombieri-Lang conjecture).

But are there some non-trivial solutions for $n\ge 5$ (and $n=4$)? Do we know if there are finitely many solutions for $n\ge 5$ ?

I guess that it should be classical, but I did not find it on this site or online, after googling "Fermat, surface, rational solutions", etc

More elliptic curves for $x^4+y^4+z^4 = 1$" in this post. $\endgroup$ – Tito Piezas III Dec 15 '15 at 3:44