TITLE: Fermat's last theorem for entire functions
QUESTION [19 upvotes]: Let $f,g,h$ be entire functions, i.e., holomorphic functions on $\mathbb{C}$. Suppose $f^n+g^n=h^n$ for some integer $n\geq2$. What can we say about $f,g,h$?

Clearly this is Fermat's last theorem for entire functions. I did a little search on the internet but, somewhat surprisingly, I found nothing relevant. Where can I find the answer? Thanks in advance. :)
Edit: In particular, I would like to know why there are no nontrivial solutions for $n\geq4$. Here a trivial solution is a solution of the form $f=ap,g=bp,h=cp$ where $a,b,c\in\mathbb{C}$ satisfy $a^n+b^n=c^n$ and $p$ is entire.

REPLY [10 votes]: Your question is equivalent to asking for non-constant meromorphic functions $f, g$ in $\Bbb C$ satisfying
$$
 f^n + g^n = 1
$$
for some integer $n \ge 2$. For $n=2$ and $n=3$ all solutions of the equation are known, see for example 

Baker, I. N. “On a Class of Meromorphic Functions.” Proceedings of the American Mathematical Society, vol. 17, no. 4, 1966, pp. 819–822. JSTOR, https://www.jstor.org/stable/2036259.

or Fermat like equation for meromorphic functions. on this site.
There are no solutions for $n \ge 4$, this goes back to 

Gross, Fred. On the equation $f^n + g^n = 1$. Bull. Amer. Math. Soc. 72 (1966), 86--88. https://projecteuclid.org/euclid.bams/1183527448

There is also a short proof using the “Second fundamental theorem” of the Nevanlinna theory: Write the equation $f^n + g^n = 1$ as
$$
 \prod_{k=1}^n \left( \frac f g - a_k \right) = \frac{1}{g^n}
$$
where $a_1, \ldots a_n$ are the $n$-th roots of $(-1)$. We can assume that $F=f/g$ is not constant (otherwise we have a “trivial” solution). Zeros of $F- a_k$ can only occur at poles of $g$, and are therefore of multiplicity $n$ or more.
Using the notation of the Nevanlinna theory this implies
$$
 \overline{N}(r, a_k, F) \le \frac 1n N(r, a_k, F) \, .
$$
Substituting this in the second fundamental theorem gives
$$
\begin{align}
 (n-2) T(r, F) &\le \sum_{k=1}^n \overline{N}(r, a_k, F) + S(r, F) \\
  &\le \frac 1n \sum_{k=1}^n N(r, a_k, F) + S(r, F) \\
 &\le T(r, F) + S(r, F)
\end{align}
$$
or $(n-3) T(r, F) \le S(r, F)$. This is a contradiction because the “error term” $S(r, F)$ is small compared to $T(r, F)$.