Primes of form Power of Two plus One/Proof 1

Proof
Suppose $n$ has an odd divisor apart from $1$.

Then $n$ can be expressed as $n = \paren {2 r + 1} s$.

So:

and so $2^n + 1$ is not prime.

Hence $2^n + 1$ can be prime only if $n$ has only even divisors.

That is, if $n = 2^k$ for some natural number $k$.