Properties of prime mod $3$ We know that if $p$ is a prime congruent to $3 \mod 4$, we cannot represent it as sum of two squares. Is there a positive property of such $p$? That is, do we have any statements that say "$p$ is a prime congruent to $3 \mod 4$ iff $\underline{\mbox{a positive statement}}$ is TRUE in an unique way". For instance, we have "$p$ is a prime congruent to $1 \mod 4$ iff $p=a^2+b^2$ and $|ab|>1$ in an unique way.

For example, we can say, a prime is $1,3 \pmod 8$ if and only if there is just one expression $p = x^2 + 2 y^2.$
You get a little flexibility by throwing in indefinite forms: a prime is $1,7 \pmod 8$ if and only if there are just two infinite sequences of expressions $p = x^2 - 2 y^2,$ under the action (and its inverse)
$$   
\left(
\begin{array}{r}
x \\
y
\end{array}
\right) \mapsto
\left(
\begin{array}{rr}
3 & 4 \\
2 & 3
\end{array}
\right)
\left(
\begin{array}{r}
x \\
y
\end{array}
\right)
$$
The two orbits for the prime $7$ have base points
$$
\left(
\begin{array}{r}
3 \\
-1
\end{array}
\right)
$$
and
$$
\left(
\begin{array}{r}
3 \\
1
\end{array}
\right)
$$
