Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$ 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
 A: For the equation.
$$x^5+y^5+z^5=w^5$$
You can write such a simple solution.
$$x=(p^2+s^2)\sqrt{-1}+p^2+2ps-s^2$$
$$y=(p^2+s^2)\sqrt{-1}+s^2-2ps-p^2$$
$$z=p^2-2ps-s^2-(p^2+s^2)\sqrt{-1}$$
$$w=(p^2+s^2)\sqrt{-1}+p^2-2ps-s^2$$
A: In the year 1988 Noam Elkies found $2682440^4+15365639^4+18796760^4=20615673^4$.
A: It has been conjectured by Euler that this equation has no solutions in positive integers when $n\geq 4$.
When $n=4$, this was disproved by Elkies in the paper [Elkies, On A4+B4+C4=D4] in a very strong way: he proves that the rational points of this K3 surface are dense in the real points for the euclidean topology.
When $n\geq 5$ is odd, your surface contains lines, for instance the line $Z-T=X+Y=0$. Consequently, it has infinitely many rational points. Of course, this does not disprove Euler's conjecture, that required positive integers.
A: I found a parameterized solution of $X^5+Y^5+Z^5=T^5$ with $X$ and $T$ rational and $Y$ and $Z$ complex rational:
$(k^2-4k+1)^5+(2k-2+(k^2-2k+3)i)^5+(2k-2-(k^2-2k+3)i)^5=(k^2-3)^5$
There is an almost Desboves form of it, as follows:
$(b-a)^5+(a+ci)^5+(a-ci)^5=(b+a)^5$, where $2a^2+b^2=c^2$
A: Here are some references:
MR2077618 Gundersen, Gary G.; Tohge, Kazuya Entire and meromorphic solutions of $f^5+g^5+h^5=1.$ Symposium on Complex Differential and Functional Equations, 57–67, Univ. Joensuu Dept. Math. Rep. Ser., 6, Univ. Joensuu, Joensuu, 2004.
https://www.researchgate.net/publication/260518318_Entire_and_meromorphic_solutions_of_f5_g5_h5_1
MR1821651 Gundersen, Gary G. Meromorphic solutions of $f^5+g^5+h^5≡1.$ Complex Variables Theory Appl. 43 (2001), no. 3-4, 293–298. 
MR1660942 Gundersen, Gary G. Meromorphic solutions of $f^6+g^6+h^6≡1.$ Analysis (Munich) 18 (1998), no. 3, 285–290. 
They study not only rational but also entire and meromorphic in the plane solutions, and they mention in these papers what is known on the subject.
There is also a survey:
MR3170744  Hayman, W. K. Waring's theorem and the super Fermat problem for numbers and functions. Complex Var. Elliptic Equ. 59 (2014), no. 1, 85–90.
According to the very recent preprint of Gundersen,
http://arxiv.org/abs/1509.02225
Equation $f^n+g^n+h^n=1$ has solutions in rational functions for $n\leq 5$ and no such solutions for $n\geq 8$. The cases $n=6, n=7$ are open. Non-constant rational solutions for $n=5$ are in the first article above.
