For any integer $\displaystyle n \geq 2$, let $\displaystyle F_n = 2^{2^n}+1$ and let $\displaystyle p | F_n$ be any prime.

I can show that $\displaystyle ord_p{2} = 2^{n+1} $, however I'm struggling to answer the rest of the qestion:

1) Using Euler’s criterion show that $\displaystyle 2^{(p-1)/2} = 1$ mod p. It looks like we have to show $\displaystyle (\tfrac{{2}}{p})$ = 1, but how can we show this?

2) Deduce that $\displaystyle p = 1 + 2^{2+n}k$ for some $\displaystyle k \in N$ and use this to verfiy that $\displaystyle F_4$ is prime.