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 over $\mathbb{Q}$ (take two different lines in $\mathbb{P}^3$ and choose one point on each one, then you get a third point through the line passing through the two points). 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

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