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arXiv · 1908.01752

On the negative Pell equation

Stephanie Chan·Peter Koymans·Djordjo Milovic·Carlo Pagano
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Abstract

Using a recent breakthrough of Smith, we improve the results of Fouvry and Kl\"uners 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 $x^2-Dy^2=-1$ 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 $\mathrm{CL}^+(D)$ of $\mathbb{Q}(\sqrt{D})$, we prove that the infimum of this density is at least $53.8\%$.

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Stephanie Chan, Peter Koymans, Djordjo Milovic, Carlo Pagano. 2019-08-05. On the negative Pell equation. https://doi.org/10.1017/fms.2022.40

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