Let $f$ and $g$ be entire functions such that $f^n+g^n=1$, where $n\geq 3$ is an integer. Prove that $f$ and $g$ are constant.
I suppose I should somehow prove that either $f$ or $g$ is bounded so that I can apply Liouville's Theorem, but I don't see how. I tried setting the derivative of the left hand side equal to zero and work with that but that did not seem to work.
Here's a proof adapted from Remmert's book Classical Topics in Complex Function Theory, page 236.
Suppose $g\neq 0$. Since $f$ and $g$ cannot have common zeros, $f/g$ is a meromorphic function that takes the value $w$ at $z$ if and only if $f(z)=wg(z)$.
We can factor the given equation as
$$1=\prod_1^n (f-\zeta_ig),$$
where the $\zeta_i$ are roots of $x^n+1$. Dividing through by $g$, we see $f/g$ cannot take the (distinct) values $\zeta_i$. By Picard's theorem for meromorphic functions, a meromorphic function that omits $3$ values is constant. So $f/g$ is constant, $f=cg$ for a constant $c$, and the rest follows easily.
