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Math Help - Legendre Symbol and Quadratic Reciprocity

  1. #1
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    Legendre Symbol and Quadratic Reciprocity

    The question asks to define a Legendre symbol and use the Law of Quadratic Reciprocity to determine whether 135 is a quadratic residue modulo 283.

    I already know how to define a Legendre symbol but I'm unsure of how to use the Law of Quadratic Reciprocity to answer this question?
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  2. #2
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    Quote Originally Posted by Pythagoras_barrel View Post
    The question asks to define a Legendre symbol and use the Law of Quadratic Reciprocity to determine whether 135 is a quadratic residue modulo 283.

    I already know how to define a Legendre symbol but I'm unsure of how to use the Law of Quadratic Reciprocity to answer this question?
    I really don't know what you mean by "defining a Legendre symbol", but since 3=283=3\!\!\!\pmod 4 , we obtain by the multiplicativity of the Legendre symbol and by the quadratic reciprocity theorem \binom{135}{283}=\binom{27}{283}\binom{5}{283} . Now:

    ** \binom{5}{283}=\binom{283}{5}=\binom{3}{5}=-1

    Tonio

    ** \binom{3}{283}=-\binom{283}{3}=-\binom{1}{3}=-1 , and then 27=3^2\cdot 3 is the product of a quadratic residue (9) and a non-quadratic residue (3), and thus it itself is a

    non-quad. res., so \binom{27}{283}=-1

    In short, 135 is the product of two non-quad. res. and thus it is a quad. residue. After some calculation, 237^2=135\!\!\!\pmod{283}
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