Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. It is easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). We suppose $D$ is not a square integer.
Let $m$ be an integer. If $m = ax^2 + bxy + cy^2$ has a solution in $\mathbb{Z}^2$, we say $m$ is represented by $F$. Suppose gcd($m, D) = 1$. Let $p$ be an odd prime divisor of $D$. By this question, $\left(\frac{m}{p}\right)$ does not depend on the choice of $m$. So it is natural to ask what can be said for the prime $2$ if $D \equiv 0$ (mod $4$).
We suppose $D \equiv 0$ (mod $4$). Let $m, k$ be odd integers represented by $f$. We would like to investigate relations between the residue class of $m$ modulo $8$ and that of $k$.
Question Are the following statements true?
1) If $D/4 \equiv 0$ (mod $8$), $mk \equiv 1$ (mod $8$).
2) If $D/4 \equiv 1, 5$ (mod $8$), $mk \equiv 1, 3, 5, 7$ (mod $8$).
3) If $D/4 \equiv 2$ (mod $8$), $mk \equiv 1, 7$ (mod $8$).
4) If $D/4 \equiv 3, 4, 7$ (mod $8$), $mk \equiv 1, 5$ (mod $8$).
5) If $D/4 \equiv 6$ (mod 8), $mk \equiv 1, 3$ (mod $8$).