  In this paper we obtain an asymptotic formula for the number of
$\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite
binary quadratic forms over $\mathbb{Z}$ having bounded discriminant $\Delta =
1-4p$, with $p$ a prime. This extends work of the first named author and has an
application in counting simple $(4a+1)$-knots of genus one.
