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Wilson's Theorem Question

  1. Nov 3, 2008 #1
    [tex]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)[/tex] mod [tex]p[/tex]
    [tex](p-1)!\equiv-1[/tex] mod p [Wilson's Theorem]
    to prove
    [tex]1^2\cdot3^2\cdot5^2\cdot...\cdot(p-2)^2\equiv(-1)^{\frac{(p-1)}{2}}[/tex] mod [tex]p[/tex]

    Relevant equations

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

    The attempt at a solution
    need assistance

  2. jcsd
  3. Nov 3, 2008 #2


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    [tex]2 \cdot 4 \cdot \ldots \cdot (p-1)=\frac{(p-1)!}{1 \cdot 3 \cdot \ldots \cdot (p-2)}=(p-1)! \cdot \left( \frac{1}{1 \cdot 3 \cdot \ldots \cdot (p-2)} \right)[/tex]
  4. Nov 3, 2008 #3
    Thanks, this problem is solved.
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