Rational solutions of the Fermat equation $X^n+Y^n+Z^n=1$

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.

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.

For the equation.

$$x^5+y^5+z^5=w^5$$

You can write such a simple solution.

$$x=(p^2+s^2)\sqrt{-1}+p^2+2ps-s^2$$

$$y=(p^2+s^2)\sqrt{-1}+s^2-2ps-p^2$$

$$z=p^2-2ps-s^2-(p^2+s^2)\sqrt{-1}$$

$$w=(p^2+s^2)\sqrt{-1}+p^2-2ps-s^2$$