For each negative discriminant, there are only finitely many proper-equivalence classes of positive-definite integral binary quadratic forms

By [L1], it is enough to show that only finitely many reduced forms have discriminant $\mathrm{\Delta }$.

If $(a,b,c)$ is reduced with discriminant $\mathrm{\Delta }$, then $1\le a\le \sqrt{\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\mathrm{/}3}$ by [L2], so only finitely many positive integers $a$ can occur.

For each such $a$, reducedness gives $\mathrm{\mid }b\mathrm{\mid }\le a$, so only finitely many integers $b$ can occur. Once $a$ and $b$ are fixed, the discriminant equation $\mathrm{\Delta }=b^2-4ac$ determines $c=(b^2-\mathrm{\Delta })\mathrm{/}(4a)$ uniquely. Therefore only finitely many reduced triples $(a,b,c)$ have discriminant $\mathrm{\Delta }$.

Hence there are only finitely many proper-equivalence classes of positive-definite integral binary quadratic forms of discriminant $\mathrm{\Delta }$.