Find all primes different from three for which $(3|p)=1$

Question

Find all primes different from three for which

$(3|p)=1$, where

$(3|p)$ denotes the Ligendre symbol.

Answer 1

Suppose $p\equiv 1 \mod 4$. Then quadratic reciprocity says $(3|p)=(p|3)$ and it's equal to $1$ iff $p\equiv 1\mod 3$. It follows that $p\equiv 1 \mod 12$.

Suppose $p\equiv -1 \mod 4$. Then quadratic reciprocity says $(3|p)=-(p|3)$ and it's equal to $1$ iff $p\equiv -1\mod 3$. It follows that $p\equiv -1 \mod 12$.

So primes you are looking for are ones of the form $12k\pm 1$.

Answer 2

By quadratic reciprocity we have $(3/p)=(p/3)$ if and only if $p\equiv 1\bmod 4$, and $(3/p)=-(p/3)$ if and only if $p\equiv 3\bmod 4$. Then $(p/3)=1$ if $p\equiv 1\bmod 3$, and $(p/3)=-1$ if $p\equiv 2\bmod 3$. This means $(3/p)=1$ if either $p\equiv 1\bmod 3$, $p\equiv 1 \bmod 4$, or $p\equiv 2\bmod 3$, $p\equiv 3 \bmod 4$. In other words, $(p/3)=1$ if and only if $p\equiv \pm 1\bmod 12$.