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 $r^k$ is a nonquadratic residue for every odd positive integer k.

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.

2. Originally Posted by Janu42
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 $r^k$ is a nonquadratic residue for every odd positive integer k.

If $\left(\frac{r}{p}\right)=1$ then $r=x^2$ . Get now a contradiction rising to the $\frac{p-1}{2}$ power both sides .

The second part follows at once

Tonio

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.
.