Properties of prime mod $3$

Question

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.

Answer 1

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 $$ \left( \begin{array}{r} 3 \\ 1 \end{array} \right) $$

Answer 2

I think I will leave the part about squareclasses alone.

Here is a diagram, of a design due to Conway, that organizes all column vectors $$ \left( \begin{array}{r} x \\ y \end{array} \right) $$ of integers $x,y$ such that $x^2 - 2 y^2 = 1,-1,2,-2,7.$

Conway's book can be downloaded from PDF. There is also a helpful discussion in Stillwell, Elements of Number Theory, particularly pages 87-99, that does a good job of showing how the value of the quadratic form and the $(x,y)$ points with $\gcd(x,y)=1$ match up, as I have illustrated for $x^2 - 2 y^2.$