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Math Help - Quadratic Residue/Legendre Symbol

  1. #1
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    Quadratic Residue/Legendre Symbol

    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.
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    Quote Originally Posted by Janu42 View Post
    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.
    .
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