Fermat Number is not Square/Proof 2

Proof
Recall the definition of Fermat numbers:


 * $F_n = 2^{(2^n)}+1$, where $n = 0, 1, 2, \ldots$

Marginal Case
$F_0 = 3$ is not a square.

General Case
It will be shown that Fermat numbers lie between $2$ consecutive squares, thus cannot be itself a square: