Class number of a negative discriminant

For a negative discriminant $D\equiv 0$ or $1(\mathrm{mod}4)$, the class number $h(D)$ is the number of proper equivalence classes of primitive positive definite integral binary quadratic forms of discriminant $D$. Equivalently, it is the number of reduced primitive forms of discriminant $D$.

For an odd prime number $q$, the primitive positive definite formshave discriminant $-8q$ and lie in distinct proper equivalence classes. The first represents one, whereas the second has minimum positive value two.

If $q\equiv 3(\mathrm{mod}8)$ is prime and $h(-8q)=2$, then for every prime $p>q$,for some integers $x,y$ exactly when $p\equiv \pm 1(\mathrm{mod}8)$ and $p$ is a quadratic residue modulo $q$.

Let $d$ be odd and have $k$ distinct prime divisors. Assigning each complete prime-power factor of $d$ to one of two factors gives $2^k$ ordered coprime factorizations $d=ac$. Interchanging $a$ and $c$ identifies pairs, and each unordered factorization gives a different reduced primitive form $[\min (a,c),0,\max (a,c)]$ of discriminant $-4d$. Hence