Results 1 to 3 of 3

Math Help - Legendre symbol

  1. #1
    Newbie
    Joined
    Sep 2011
    Posts
    3

    Legendre symbol

    Hey guys, I need a bit of help here.

    1. Show that the equation x^2-71y^2=19 has no integer solution using Legendre symbol.

    So \left(\frac{a}{p}\right) \equiv a^{(p-1)/2}\ \pmod{p}

    \left(\frac{71}{19}\right) \equiv 71^{(19-1)/2}\ \pmod{19}

    71^9\ \pmod{19} = 18 = -1

    So because it equals -1, 71 is a quadratic non-residue modulo 19 and so there is no integer solution to x^2-71y^2=19.

    Is this enough to show no integer solutions?


    2. Evaluate the Legendre symbol \left(\frac{p^2-1}{p}\right) for odd prime p.

    So \left(\frac{p^2-1}{p}\right) \equiv (p^2-1)^{(p-1)/2}\ \pmod{p}

    What is the formal way to go from here?
    I have evaluated this for several cases of odd prime p and kind of see a pattern.

    Also, this value alternates between -1 and 1 for odd numbers.

    Any help guys?

    Thanks for your help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Re: Legendre symbol

    1. x^2-71y^2=19 \implies x^2\equiv 19 \pmod{71}

    \left(\frac{19}{71}\right)= \left(\frac{71}{19}\right)=-1

    So, 19 is a quadratic non-residue modulo 71 and therefore the equation has no integer solutions.

    2. \left(\frac{p^2-1}{p}\right)=\left(\frac{(p+1)(p-1)}{p}\right)

    =\left(\frac{1}{p}\right)\left(\frac{-1}{p}\right)

    =(-1)^{(p-1)/2}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2011
    Posts
    3

    Re: Legendre symbol

    Oh cool, thanks alex, I understand it!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Legendre symbol
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: May 16th 2011, 08:55 PM
  2. [SOLVED] Legendre Symbol
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: October 31st 2010, 10:01 AM
  3. Legendre Symbol (2/p)
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: March 9th 2010, 08:46 PM
  4. the legendre symbol
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 3rd 2008, 03:34 PM
  5. Legendre symbol
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: May 7th 2008, 09:36 AM

Search Tags


/mathhelpforum @mathhelpforum