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Math Help - hello to all! need help in number theory

  1. #1
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    hello to all! need help in number theory

    Let p be a prime of the form 4k+3. Prove that either



    {(p-1)/2}!≡1 (mod p) or{(p-1)/2}!≡-1 (mod p)
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  2. #2
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    Quote Originally Posted by jen_mojic View Post
    Let p be a prime of the form 4k+3. Prove that either



    {(p-1)/2}!≡1 (mod p) or{(p-1)/2}!≡-1 (mod p)
    By Wilson's theorem we have,
    1\cdot 2\cdot 3 \cdot ... \cdot (p-1) \equiv -1(\bmod p).
    Now, p-1 \equiv -1, and p-2\equiv -2, and so on ... until the middle.
    Thus, (-1)^{(p-1)/2} \left[ (\tfrac{p-1}{2})! \right] \equiv -1(\bmod p).
    However, (-1)^{(p-1)/2} = -1 since p=4k+3.
    And therefore we have, \left[(\tfrac{p-1}{2})!\right]^2 \equiv 1(\bmod p) \implies (\tfrac{p-1}{2})! \equiv \pm 1(\bmod p).

    For example, let p=7, then,
    1\cdot 2\cdot 3 \cdot 4\cdot 5 \cdot 6 \equiv -1(\bmod 7)
    Do the trick above,
    1\cdot 2 \cdot 3 \cdot (-3) \cdot (-2)\cdot (-1) \equiv -1(\bmod 7)
    Thus,
    (-1)^3 (3!)^2 \equiv -1(\bmod 7)
    And the rest follows.
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