On the negative Pell equation

@article{Chan2019OnTN,
  title={On the negative Pell equation},
  author={Stephanie Chan and Peter Koymans and Djordjo Z. Milovic and Carlo Pagano},
  journal={arXiv: Number Theory},
  year={2019}
}
Using a recent breakthrough of Smith, we improve the results of Fouvry and Kluners 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… 

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