Question:
Prove: If p and q=2p+1 are both odd primes, then -4 is a primitive root of q.
Note that when then (1) . There are primitive roots and non-quadratic residues, and each primitive root is a non-quadratic residue (because otherwise it'd only generate quadratic residues). Thus (1) means that all non-quadratic residues but 1 of them, are primitive roots.
We can show that the only non-quadratic residue that is not a primitive root is -1. Note that (consider p=1,3(mod.4) ), thus -1 is a non-quadratic residue mod. q. But thus -1 is not a primitive root.
Now just check that is a non-quadratic residue module q (easy because 4 is a square and -1 is a non-quadratic residue), and then check that and we are done.