I think the proof would be a lot easier if you was an odd prime. Anyway, if is a primitive root then,
Contains all the elements of
But since is a quadradic residue.
That means is a quadradic residue.
Then all numbers are quadradic residues of , which is surly not possible unless .
Note: By odd primes the proof is extremely simple if is a quadradic residue then,
But that is not possible because the order of is .