Coprime-factorization lower bound for a quadratic-form class number
= Coprime-factorization lower bound for a quadratic-form class number
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
$$
h(-4d)\geq2^{k-1}.
$$