## Find all odd primes p for which (11/p) = 1. Quadratic residue

Find all odd primes $p$ for which $\left(\frac{11}{p}\right) = 1$. $\left(\frac{11}{p}\right)$ is in Legendre Symbol.

$\left(\frac{11}{p}\right) = - \left( \frac{p}{11} \right)$ due to quadratic reciprocity. ( $11 \equiv 3 \pmod{4}$)

Since $11$ is prime and $p$ is prime, we have $gcd(11, p) = 1$.

Apply Gauss' theorem.

$\frac{11 - 1}{2} = 5$.
Let $S = \{p, 2p, 3p, 4p, 5p \}$.
Let $k = \{ s | s > 5 \}$
Then $\left(\frac{p}{11}\right) = (-1)^k = -1$ iff $k$ is odd.

When $p = 3$, $k = 4$.
When $p = 5$, $k = 4$.
For all odd primes $p > 5$, $k = 5$.

So all odd primes $p > 5$ satisfy $\left(\frac{11}{p}\right) = 1$.

Is my attempt correct?