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Thread: Probability of a square Mod

  1. #1
    Dec 2009

    Probability of a square Mod

    I really have no idea where to start , or how to do this problem.

    Suppose x is relatively prime to p; it's claimed when p is large that x is a square root mod p with probability of 1/2.

    I'm supposed to justify the claim.

    If anyone could help me with this it would be greatly appreciated.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
    Jun 2009
    You say that $\displaystyle x$ is a square root mod $\displaystyle p$? That is true all of the time. $\displaystyle x$ is the square root of $\displaystyle x^2$, mod $\displaystyle p$.

    What you probably mean is that $\displaystyle x$ is a square mod $\displaystyle p$ with probability 1/2. This is because there are as many nonzero squares as there are non-squares mod $\displaystyle p$ (when $\displaystyle p$ is odd).

    To see that this is true, notice that the nonzero squares mod $\displaystyle p$ are precisely the roots of the polynomial $\displaystyle x^{(p-1)/2}-1$, which has no repeated roots.
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