Is there any number that satisfies $\lvert n\rvert > 2, n\in\Bbb{C}$ and $a^n+b^n=c^n$ where $a,b,c \in \Bbb{C}$

Question

Fermat's Last Theorem has been proven, and this means that there are no integer values of $n$ that satisfy the equation $a^n+b^n=c^n$ where $a,b,c \in \Bbb{C}$ for $n>2$. But are there any rational values of $n$ with solutions? Any real values? Complex values? Has 2 been proven to be the largest $n$ with solutions?

Answer 1

Fermat's Last Theorem rules out rational values for $n \gt 2$, because you can clear the fractions and get an integral solution. In the reals there are solutions for all $n$. For example $3^4+4^4=337=(\sqrt[4]{337})^4$