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

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