Assume a primitive root modulo is a QR. Then there exists an such that . But then . Hence we just showed has an order less than . Therefore we've reached a contradiction.
Let p be an odd prime. Prove that every primitive root of p is a quadratic nonresidue. Prove that every quadratic nonresidue is a primitive root if and only if is of the form where is a non-negative integer, that is, if and only if is a Fermat number.
I have no clue what to do.