On the number of binary quadratic forms having discriminant $1-4p$, $p$ prime

Abstract: 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.