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Math Help - Quad Reciprocity

  1. #1
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    Quad Reciprocity

    Establish the following assertion: (5/p)=1 if and only if p=1, 9, 11, or 19(mod 20)
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  2. #2
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    Quote Originally Posted by cathwelch View Post
    Establish the following assertion: (5/p)=1 if and only if p=1, 9, 11, or 19(mod 20)
    \left ( \frac{5}{p} \right ) = \left ( \frac{p}{5} \right ) (-1)^{(p-1)(5-1)/4} = \left ( \frac{p}{5} \right )

    Therefore p \equiv 1, 4 \pmod{5}

    Therefore p \equiv 1, 4, 6,9, 11,14,16,19  \pmod{20}

    However, p is odd, therefore

    p \equiv 1, 9, 11, 19 \pmod{20}
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  3. #3
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    thanks, but did you use some sort of property or theorem
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  4. #4
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    Never mind, I see that you used the Quadratic Reciprocity Law
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