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Thread: proof with quadratic residue

  1. #1
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    proof with quadratic residue

    Prove that if $\displaystyle p \equiv 3 (\mod{8})$ and $\displaystyle (p-1)/2$ is prime then $\displaystyle (p-1)/2$ is a quadratic residue $\displaystyle (\mod{p})$.

    I believe I need to show that $\displaystyle (((p-1)/2)/p) = 1$. Will someone show me where to start? I have been thinking about this problem all weekend.

    Thanks in advance,
    Last edited by Zalren; Jul 17th 2011 at 01:03 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Re: proof with quadratic residue

    Well, if $\displaystyle p \equiv 3 \mod p$, that doesn't leave you much choice for the value of $\displaystyle p$, does it?
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  3. #3
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    Re: proof with quadratic residue

    Oops, I typed it wrong. I corrected it!
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  4. #4
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    Re: proof with quadratic residue

    What if I started like this:

    Using Quadratic Reciprocity Theorem: $\displaystyle (((p-1)/2)/p)(p/((p-1)/2)) = (-1)^{((((p-1)/2)-1)/2)((p-1)/2)} = 1$ since $\displaystyle (p-1)/2$ is an even exponent. That tells me that $\displaystyle (((p-1)/2)/p)=(p/((p-1)/2))$

    $\displaystyle p \equiv 1 (\mod{(p-1)/2})$, I believe this is a true statement, how do I back it up?

    So $\displaystyle (((p-1)/2)/p)=(p/((p-1)/2))=(1/((p-1)/2))=1$
    Last edited by Zalren; Jul 17th 2011 at 01:44 PM.
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    MHF Contributor Bruno J.'s Avatar
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    Re: proof with quadratic residue

    $\displaystyle p=2\times ((p-1)/2)+1$.

    And $\displaystyle (p-1)/2$ is not even if $\displaystyle p \equiv 3 \mod 8$...
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  6. #6
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    Re: proof with quadratic residue

    So can I assume $\displaystyle p>3$, for example $\displaystyle p = 11, 19, ...$, and just show the specific case where $\displaystyle p=3$ and show that $\displaystyle (((p-1)/2)/p) = (((3-1)/2)/3) = (1/3) = 1$?
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