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Math Help - Legendre symbol

  1. #1
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    Legendre symbol

    Calculate the legendre symbol using quadratic reciprocity \frac{31}{43}

    I have calculated this and found it to be -1.
    I am wondering if there is a way to check my answer. I tried to check on scientific notebook but that does not seem to have a function for the legendre symbol and i cannot find any online software for this.
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  2. #2
    Super Member Deadstar's Avatar
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    Maple.

    with(numtheory):

    legendre(31,43);

    I found the answer to be 1 though...

    \bigg{(} \frac{31}{43} \bigg{)} = \bigg{(} \frac{43}{31} \bigg{)} (-1)^{\tfrac{31-1}{2} \cdot \tfrac{43-1}{2}} = -\bigg{(} \frac{43}{31} \bigg{)}

    = -\bigg{(} \frac{12}{31} \bigg{)}

    = -\bigg{(} \frac{2^2}{31} \bigg{)} \cdot \bigg{(} \frac{3}{31} \bigg{)}

     = -\bigg{(} \frac{3}{31} \bigg{)}

     = -\bigg{(} \frac{31}{3} \bigg{)} (-1)^{\tfrac{31-1}{2} \cdot \tfrac{3-1}{2}}

     = \bigg{(} \frac{31}{3} \bigg{)}

     = \bigg{(} \frac{1}{3} \bigg{)} = 1
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  3. #3
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    Quote Originally Posted by Deadstar View Post
    Maple.

    with(numtheory):

    legendre(31,43);

    I found the answer to be 1 though...

    \bigg{(} \frac{31}{43} \bigg{)} = \bigg{(} \frac{43}{31} \bigg{)} (-1)^{\tfrac{31-1}{2} \cdot \tfrac{43-1}{2}} = -\bigg{(} \frac{43}{31} \bigg{)}

    = -\bigg{(} \frac{12}{31} \bigg{)}

    = -\bigg{(} \frac{2^2}{31} \bigg{)} \cdot \bigg{(} \frac{3}{31} \bigg{)}

     = -\bigg{(} \frac{3}{31} \bigg{)}

     = -\bigg{(} \frac{31}{3} \bigg{)} (-1)^{\tfrac{31-1}{2} \cdot \tfrac{3-1}{2}}

     = \bigg{(} \frac{31}{3} \bigg{)}

     = \bigg{(} \frac{1}{3} \bigg{)} = 1

    I got the same and in fact 31=17^2\!\!\!\pmod {43}

    Tonio
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  4. #4
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    i had split 12/31
    into 6 lots of 2/31 and then used the property that p is a prime of the form 8k-1 and thus 2/31 =1. Which works out the same except that i had forgotten the negative sighn from the first calculation of 31/43=-(43/31)

    Thaks for the help though this method seems a bit easier.
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