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Proving there are infinitely many primes

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that their are infinitely many primes such that [itex]\left(\frac{14}{p}\right)=1[/itex]

    2. Relevant equations

    The bracketed symbol is the legendre symbol (i.e. there are infinitely many primes such that 14 is a square modulo p)

    3. The attempt at a solution

    Well by quadratic reciprocity [itex]\left(\frac{14}{p}\right)=\left(\frac{2}{p}\right)\left(\frac{7}{p}\right)= (-1)^{\frac{p^{2}-1}{2}}(-1)^{\frac{p-1}{2}\frac{7-1}{2}} \left(\frac{p}{7}\right)[/itex].

    The first part, [itex](-1)^{\frac{p^{2}-1}{2}}[/itex] is 1 if [itex]p=1,7(mod 8)[/itex] or -1 if [itex]p=3,5(mod8)[/itex].

    But I'm not sure how I can use that fact on the remainder of the equation, which would require modulo 7.
     
  2. jcsd
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