Primes of form Power of Two plus One/Proof 1

Theorem
Let $n \in \N$ be a natural number.

Let $2^n + 1$ be prime.

Then $n = 2^k$ for some natural number $k$.

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

Then $n$ can be expressed as $n = \left({2 r + 1}\right) s$.

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 $k$.