# Primes of form Power of Two plus One/Proof 2

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## 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

A specific instance of Primes of form Power plus One:

$q^n + 1$ is prime only if:

- $(1): \quad q$ is even

and

- $(2): \quad n$ is of the form $2^k$ for some positive integer $k$.

As $2$ is even, the result applies.

$\blacksquare$