# Homework Help: Proving there are infinitely many primes

1. Apr 22, 2012

### SP90

1. The problem statement, all variables and given/known data

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

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 $\left(\frac{14}{p}\right)=\left(\frac{2}{p}\right)\left(\frac{7}{p}\right)= (-1)^{\frac{p^{2}-1}{2}}(-1)^{\frac{p-1}{2}\frac{7-1}{2}} \left(\frac{p}{7}\right)$.

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

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