Fermat Number is not Square/Proof 1

Proof
Let $n = 0$.

Then $F_0 = 2^{2^0} + 1 = 3$ is not a square.

Let $n \ge 1$.

Then:

Thus by Zero and One are the only Consecutive Perfect Squares, $2^{\left({2^n}\right)} + 1$ is not square.