1 plus Perfect Power is not Power of 2

Theorem
The equation:
 * $1 + a^n = 2^m$

has no solutions in the integers for $n, m > 1$.

This is an elementary special case of Catalan's Conjecture.

Proof
there is a solution.

Then:

$a$ is immediately seen to be odd.

By Square Modulo 4, $n$ must also be odd.

Now:

The latter sum has $n$ powers of $a$, which sums to an odd number.

The only odd divisor of $2^m$ is $1$.

However, if the sum is $1$, we have:
 * $a^n + 1 = a + 1$

giving $n = 1$, contradicting our constraint $n > 1$.

Hence the result by Proof by Contradiction.