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Math Help - Congruence Problems

  1. #1
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    Congruence Problems

    a) Let p be an odd prime and let
    a = \prod (2j-1) = (1)(3)(5)....(p-2)
    Prove that a^2 = (-1)^{(p+1)/2} mod p

    (Product is from j=1 to (p-1)/2)

    b) Let p be a prime. Prove that
    \left(\begin{array}{cc}2p\\p\end{array}\right) = 2 mod p

    \left(\begin{array}{cc}2p\\p\end{array}\right) is 2p choose p, as in statistics. so (2p)!/p!(2p-p)!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hint for a:

    Wilson theorem.
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  3. #3
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Hint for a:

    Wilson theorem.
    Wilson's crossed my mind but I don't see how to work (p-1)! into the proof
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Another hint(or solution)...

    t\equiv -(p-t)(\mod p), so

    2\cdot4\cdot6 ... (p-1)\equiv (-1)^{\frac{p-1}{2}}1\cdot3\cdot5...(p-2)(\mod p)

    ...
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