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Math Help - Legendre Symbol

  1. #1
    Junior Member
    Joined
    May 2009
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    25

    Legendre Symbol

    Question:
    If p is an odd prime and gcd (ab, p)=1, prove that at least one of a, b, or ab is a quadratic residue of p.
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  2. #2
    Super Member PaulRS's Avatar
    Joined
    Oct 2007
    Posts
    571
    Prove it by contradiction, assume none of them were a quadratic residue, then <br />
\left( {\tfrac{a}<br />
{p}} \right) = \left( {\tfrac{b}<br />
{p}} \right) = \left( {\tfrac{{ab}}<br />
{p}} \right) =  - 1<br />
however remember that <br />
\left( {\tfrac{a}<br />
{p}} \right) \cdot \left( {\tfrac{b}<br />
{p}} \right) = \left( {\tfrac{{ab}}<br />
{p}} \right)<br />
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