Let $f=(a, b, c)$ be a positive definite binary quadratic form with integer coefficients. What does it mean to say that $f$ is reduced? Show that if $f$ is reduced and has discriminant $d$, then $|b| \leqslant a \leqslant \sqrt{|d| / 3}$ and $b \equiv d(\bmod 2)$. Deduce that for fixed $d<0$, there are only finitely many reduced $f$ of discriminant $d$.

Find all reduced positive definite binary quadratic forms of discriminant $-15$.