Fermat Number is not Perfect Power

Theorem

There exist no Fermat numbers which are perfect powers.

Proof

Each Fermat number is in the form of $1 + 2^n$ for some $n \in \Z$.

This $n$ must also be a power of $2$.

From 1 plus Power of 2 is not Perfect Power except 9 we have:

$1 + 2^n = a^b$

has only one solution $\tuple {n, a, b} = \tuple {3, 3, 2}$.

But $3$ is not a power of $2$.

Hence no Fermat numbers are perfect powers.

$\blacksquare$