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Math Help - Legendre symbol

  1. #1
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    Legendre symbol

    Determine the values of

    47/37 and

    3/43

    stating any properties of the legendre symbol used.

    is there just one method to solve all of these or are there different methods to solve depending on the numbers.
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  2. #2
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    Quote Originally Posted by Rakesh View Post
    47/37
    (47/37) = (10/37) ---> Congruence
    =(2/37)(5/37) ---> Multiplicative

    (2/37) = -1 because 37\equiv \pm 3(\bmod 8).
    (5/37) = (37/5) ---> Quadradic Reciprocity
    (37/5) = (2/5) ---> Congruence
    (2/5) = 1 because 5\equiv \pm 1(\bmod 8).

    Thus, (10/37) = (2/37)(5/37) = (-1)(1)=-1.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    (2/5) = 1 because 5\equiv \pm 1(\bmod 8).

    Thus, (10/37) = (2/37)(5/37) = (-1)(1)=-1.
    Shouldnt the above read like this?

    (2/5) = -1 because 5\equiv \pm 3(\bmod 8).

    Thus, (10/37) = (2/37)(5/37) = (-1)(-1)= 1.
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  4. #4
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    Quote Originally Posted by Isomorphism View Post
    Shouldnt the above read like this?

    (2/5) = -1 because 5\equiv \pm 3(\bmod 8).

    Thus, (10/37) = (2/37)(5/37) = (-1)(-1)= 1.

    That's what I thought, because 7 \equiv -1 \mod8 and therefore not 5.

    Also by Quadratic Reciprocity \frac {2}{5} =-1 as 2 is clearly not a square mod 5
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  5. #5
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    Quote Originally Posted by Isomorphism View Post
    Shouldnt the above read like this?

    (2/5) = -1 because 5\equiv \pm 3(\bmod 8).

    Thus, (10/37) = (2/37)(5/37) = (-1)(-1)= 1.
    Quote Originally Posted by jtsab View Post
    That's what I thought, because 7 \equiv -1 \mod8 and therefore not 5.

    Also by Quadratic Reciprocity \frac {2}{5} =-1 as 2 is clearly not a square mod 5
    Yes. It is exactly how Isomorphism said it should be.
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