.Let p be an odd prime and r be a primitive root of p.
a) Prove that the Legendre symbol (r/p) = -1 and conclude that is a nonquadratic residue for every odd positive integer k.
If then . Get now a contradiction rising to the power both sides .
The second part follows at once
b) Prove that if every nonquadratic residue of p is a primitive root of p, then p is a Fermat prime. (Hint: Part (a) should be helpful)
c) Prove that if p is a Fermat prime, then ever nonquadratic residue modulo p is a primitive root of p.