# Entire $f$ and $g$ satisfying $f^n+g^n=1$ for $n>3$ must be constant. [duplicate]

Studying for a qualifying exam and came across Don Marshall's notes. This one has stumped me for some time. This question is related to another MSE question:

\begin{align*}\text{Iff$and$g$are entire and } [f(z)]^n+[g(z)]^n=1\text{ for$n>3$then$f,gare constant}. \end{align*}

The most I could deduce is that if $f$ were not constant then $[f(z)]^n$ is necessarily surjective as missing one value for $[f(z)]^n$ would force $n$ missed values for $f$. That's about as far as I could get.

• You were very close ;) – Daniel Fischer Mar 15 '14 at 20:43
• Absolutely a duplicate. Despite my search I couldn't find the page you linked to before I posted. – user135671 Mar 15 '14 at 20:47
• No problem, searching isn't easy. I knew something more to look for, and I knew there was the duplicate. So I was able to find it. – Daniel Fischer Mar 15 '14 at 20:50