How to demonstrate that $2^{2^n - 2} + 1$ is a nonprime number? This, considering $n ≥ 3$.
I have tried by induction; I suppose that it's true for all n less than or equal to k (and greater than or equal to 3), but then I stride when I go to prove for n = k + 1.
$2^{2^{k+1} - 2 } + 1 = $
$2^{2* 2^k - 2} + 1$

Multiply by $4$ and look at things mod $5$:
$$4(2^{2^n-2}+1)=2^{2^n}+4=2^{4\cdot2^{n-2}}+4\equiv1+4\equiv0\mod5$$
so $5$ is a prime divisor of $2^{2^n-2}+1$.  Note that for $n=2$, $2^{2^2-2}+1=5$, but for $n\ge3$, $2^{2^n-2}+1\gt5$, so $5$ is a proper divisor.
