# odd primes p for which 7 is a quadratic residue mod p

• Mar 31st 2011, 05:20 AM
uberbandgeek6
odd primes p for which 7 is a quadratic residue mod p
Find all odd primes p for which 7 is a quadratic residue mod p.

I started out with the legendre symbol (7 / p)L = (-1)^(((7-1)/2)*((p-1)/2))*(p / 7)L = (-1)^(3(p-1)/2)*(p / 7)L by the quadratic reciprocity law, and I know I want it to be equal to 1. Now I'm not sure where to go from here.
• Mar 31st 2011, 12:04 PM
tonio
Quote:

Originally Posted by uberbandgeek6
Find all odd primes p for which 7 is a quadratic residue mod p.

I started out with the legendre symbol (7 / p)L = (-1)^(((7-1)/2)*((p-1)/2))*(p / 7)L = (-1)^(3(p-1)/2)*(p / 7)L by the quadratic reciprocity law, and I know I want it to be equal to 1. Now I'm not sure where to go from here.

What is that "L" you write there?? Anyway. By Gauss Quadratic Reciprocity, and since $7-1=6$ , we have that

$\displaystyle{\left(\frac{7}{p}\right)=(-1)^{\frac{p-1}{2}}\left(\frac{p}{7}\right)=\left\{\begin{array }{rl}1&\mbox{ , if }\left\{\begin{array}{} p=1\!\!\pmod 4\mbox{ and }p=1,2\mbox{ or }4\!\!\pmod 7\\p=3\!\!\pmod 4\mbox{ and }p\neq 1,2\mbox{ or }4\!\!\pmod 7\end{array}\right.\\-1&\mbox{ , otherwise}\end{array}\right.}$

Tonio