Good day! I'm currently studying on the primitive roots mod n. Eventually, I fully understand the concept of calculating the primitive roots of a number by practice, but I encounter the following problems that is out of my league. Please help me understand the following concept. Any idea will be of great help.

1. Show that $$ Fn = 2 ^ {2 ^ n} + 1 $$ , n > 1, is a prime, then 2 is not a primitive roots of Fn.
2. Show that if p = 1 (mod 4) and $g$ is a primitive root of $p$, then so is $-g$. Show by a numerical example that this need not be the case if p = 3 (mod 4).
3. Prove that if $a$ has order hk (mod m), then $a^{k}$ has order k mod n.
4. Prove that if a has order 2k modulo p the odd prime p, then a^k = -1 (mod p).