TITLE: Show that a number of the form $2^{n}+1$ can never be an integer raised to an odd power.
QUESTION [5 upvotes]: Generally I tried analyzing cases such as:
If there is $a\in\mathbb{Z}$ such that $2^n+1=a^{m}$ where $m$ is odd, then $a$ can not be negative. But a couldn't figure out what to do next.
I also tried using the factorization: $a^{m}+1=(a+1)(a^{m-1}-a^{m-2}+\dots+1)$ that makes sense for $m$ odd. I applied it after summing $1$ to both sides of the equation.
I did get some contradictions after concluding that $a+1$ divides $2(2^{n-1}+1)$ but I couldn't analyze all cases.

REPLY [3 votes]: $a$ should odd because $2^n$ is even.
Then in the factorization $\underbrace{2^n}_\text{even}=\underbrace{(a-1)}_\text{even}\underbrace{(1+a+a^2+\cdots+a^{m-1}}_\text{m odd terms})$
And a sum of $m$ odd terms with $m$ odd cannot be even.