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Math Help - quadratic congruence

  1. #1
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    quadratic congruence

    Hi,

    I am not too good at this stuff, but getting better haha..

    Anyway, find an x which satisfies the statement 5003 divides x^2 + 10, or else prove that no such x exists.


    I started by switching it around to say x^2 = -10 mod 5003.
    I kind of suspect there is no solution, and there is a theorem (euler's criterion) which states that -10 is a quadratic residue (i.e. there is a solution) if and only if (-10)^(5002/2) = (-10)^2501 = 1 mod 5003.

    (-10)^2501 is a huge number tho, and i'm not sure how to deal with it. Any hints would be great.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    You need to use the Euler-Gauss quadratic reciprocity law, along with the fact that \left(\frac{-10}{5003}\right)=\left(\frac{-1}{5003}\right)\left(\frac{2}{5003}\right)\left(\f  rac{5}{5003}\right). (Assuming you are familiar with this notation for the Legendre symbol).
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