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Math Help - Primitive Roots

  1. #1
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    Primitive Roots

    Let p be an odd prime number and let r be an integer with p not dividing r. Prove that r is a primitive root modulo p if and only if r^((p-1)/q) is not congruent to 1 mod p for all prime divisors q of p-1.
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  2. #2
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    The order of the multiplicative group \mathbb{Z}_{p}^\times is p-1.
    The idea is that r is a primitive root if the order of r is p-1.
    The order of any element of \mathbb{Z}_{p}^\times must divide p-1.

    Do you get where this is going?
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    not really?
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    Quote Originally Posted by mndi1105 View Post
    Let p be an odd prime number and let r be an integer with p not dividing r. Prove that r is a primitive root modulo p if and only if r^((p-1)/q) is not congruent to 1 mod p for all prime divisors q of p-1.
    Let \gcd(r,p)=1. It should be a known result that if d is order of r then r^d \equiv 1(\bmod p) and d divides p-1. Therefore, if for all d>1 which divide p-1 we have r^d \not \equiv 1(\bmod p) then r must be a primitive root. That is essentially what you problem is saying.
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