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Math Help - Quad Congrnence

  1. #1
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    Quad Congrnence

    Question:

    Prove that the quadratic congruence 6x^2+5x+1=0(mod p) has a solution for every prime p, even though the equation 6x^2+5x+1=0 has no solution in the integers.
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  2. #2
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    Quote Originally Posted by cathwelch View Post
    Question:

    Prove that the quadratic congruence 6x^2+5x+1=0(mod p) has a solution for every prime p, even though the equation 6x^2+5x+1=0 has no solution in the integers.
    There is a solution x=1 if p=2 or p=3.

    The solution in the reals is x = \frac{-5\pm1}{12}. If p is not equal to 2 or 3 then 12 is coprime to p, and so 12 will have an inverse k (mod p). Then x = k(-5\pm1) will be solutions to 6x^2+5x+1=0 (mod p).
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