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Math Help - primitive roots

  1. #1
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    primitive roots

    could you please tell me the primitive roots of 83, and 149.

    also do you know a way to find out the number of primitive roots a number has, it doesn't necessarily have to tell me what they are

    thanks
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    Quote Originally Posted by hanjp123 View Post
    could you please tell me the primitive roots of 83, and 149.

    also do you know a way to find out the number of primitive roots a number has, it doesn't necessarily have to tell me what they are

    thanks
    This and this might help you.
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  3. #3
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    i'm still kind of confused
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  4. #4
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    Once you find a primitive root, then you can find all other primitive roots by using this fact:

    Let g be a primitive root modulo n. Then g^k is also a primitive root if and only if \gcd(k, \phi (n)) = 1.

    So for 89, notice that 3 is a primitive root. Then 3^k where \gcd(k, \phi (89)) = 1 occurs will give you another primitive root.

    As for the number of primitive roots, all primes have \phi (p-1) primitive roots.

    Even more general, if some n has primitive roots, then there are \phi \left( \phi (n)\right) of them.
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