On the negative Pell equation

Abstract

Using a recent breakthrough of Smith, we improve the results of Fouvry and Klüners on the solubility of the negative Pell equation. Let $\mathcal{D}$ denote the set of fundamental discriminants having no prime factors congruent to $3$ modulo $4$. Stevenhagen conjectured that the density of $D$ in $\mathcal{D}$ such that the negative Pell equation $x^2-D{y}^2=-1$ is solvable with $x,y\in \mathbb{Z}$ is $58.1\%$, to the nearest tenth of a percent. By studying the distribution of the $8$-rank of narrow class groups ${\mathrm{CL}}^{+}(D)$ of $\mathbb{Q}(\sqrt{D})$, we prove that the infimum of this density is at least $53.8\%$.