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Math Help - [SOLVED] Wilson's Theorem Question

  1. #1
    Newbie
    Joined
    Nov 2008
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    5

    [SOLVED] Wilson's Theorem Question

    Use:
    2\cdot4\cdot...\cdot(p-1)\equiv(2-p)(4-p)\cdot...\cdot(p-1-p)\equiv(-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2) mod p
    and
    (p-1)!\equiv-1 mod p [Wilson's Theorem]
    to prove
    1^2\cdot3^2\cdot5^2\cdot...\cdot(p-2)^2\equiv(-1)^{\frac{(p-1)}{2}} mod p

    Relevant equations

    Gauss lemma
    wilson's theorem [ (p-1)!\equiv-1 mod p]

    The attempt at a solution
    need assistance

    Thanks

    PROBLEM SOLVED
    Last edited by mathsss2; November 2nd 2008 at 10:36 PM.
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  2. #2
    Senior Member
    Joined
    Dec 2007
    From
    Melbourne
    Posts
    428
    I believe what you are being asked to prove is incorrect:
    2\cdot4\cdot...\cdot(p-1)\equiv(-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2)
    then multiply both sides by
    (-1)^{\frac{(p-1)}{2}}\cdot1\cdot3\cdot...\cdot(p-2)

    To get
    1^2\cdot 3^2\cdot 5^2\cdot ... (p-2)^2 \equiv (p-1)!(-1)^{\frac{(p-1)}{2}}
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