Geodesic
The 8-rank of the narrow class group and the negative Pell equation
Forum of Mathematics, Sigma, Tome 10 (2022)

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Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let $\mathcal {D}$ denote the set of positive squarefree integers having no prime factors congruent to $3$ modulo $4$. Stevenhagen [19] conjectured that the density of d in $\mathcal {D}$ such that the negative Pell equation $x^2-dy^2=-1$ 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 $\operatorname {\mathrm {Cl}}^+(d)$ of $\mathbb {Q}(\sqrt {d})$, we prove that the infimum of this density is at least $53.8\%$. 
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     title = {The 8-rank of the narrow class group and the negative {Pell} equation},
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Stephanie Chan; Peter Koymans; Djordjo Milovic; Carlo Pagano. The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.40

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