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 $x^2Dy^2=1$ 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 $53.8\%$.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.01752
 Bibcode:
 2019arXiv190801752C
 Keywords:

 Mathematics  Number Theory
 EPrint:
 Forum Math. Sigma 10 (2022), Paper No. e46