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Math Help - Primitive root

  1. #1
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    Primitive root

    somebody please help me with this one:

    If r is a primitive root of p^2, p being an odd prime. show that the solutions of the congruence X^(p-1)=1 (mod p^2) are precisely the integer r^p, r^(2p),......r^((p-1)p)

    thanks
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  2. #2
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    Quote Originally Posted by felixmcgrady View Post
    somebody please help me with this one:

    If r is a primitive root of p^2, p being an odd prime. show that the solutions of the congruence X^(p-1)=1 (mod p^2) are precisely the integer r^p, r^(2p),......r^((p-1)p)
    Check that all of these are solutions by substituion and that r^i \not \equiv r^j ~ (\bmod p^2) for i\not = j.

    Now if x is solution then x\equiv r^y and so r^{(p-1)y} \equiv 1(\bmod p^2). Therefore, (p-1)y \equiv 0 (\bmod p(p-1)).

    Now \gcd (p-1,p(p-1)) = p-1 so there are p-1 solutions to this congruence. Which means we found all of them.
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