The 8-rank of the narrow class group and the negative Pell equation

Chan S, Koymans P, Milovic D, Pagano C. 2022. The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma. 10, e46.

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Author
Chan, StephanieISTA ; Koymans, Peter; Milovic, Djordjo; Pagano, Carlo
Abstract
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 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 D such that the negative Pell equation x2−dy2=−1 is solvable with x,y∈Z is 58.1% , to the nearest tenth of a percent. By studying the distribution of the 8 -rank of narrow class groups Cl+(d) of Q(√d) , we prove that the infimum of this density is at least 53.8% .
Publishing Year
Date Published
2022-05-17
Journal Title
Forum of Mathematics, Sigma
Publisher
Cambridge University Press
Volume
10
Article Number
e46
ISSN
IST-REx-ID

Cite this

Chan S, Koymans P, Milovic D, Pagano C. The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma. 2022;10. doi:10.1017/fms.2022.40
Chan, S., Koymans, P., Milovic, D., & Pagano, C. (2022). The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma. Cambridge University Press. https://doi.org/10.1017/fms.2022.40
Chan, Stephanie, Peter Koymans, Djordjo Milovic, and Carlo Pagano. “The 8-Rank of the Narrow Class Group and the Negative Pell Equation.” Forum of Mathematics, Sigma. Cambridge University Press, 2022. https://doi.org/10.1017/fms.2022.40.
S. Chan, P. Koymans, D. Milovic, and C. Pagano, “The 8-rank of the narrow class group and the negative Pell equation,” Forum of Mathematics, Sigma, vol. 10. Cambridge University Press, 2022.
Chan S, Koymans P, Milovic D, Pagano C. 2022. The 8-rank of the narrow class group and the negative Pell equation. Forum of Mathematics, Sigma. 10, e46.
Chan, Stephanie, et al. “The 8-Rank of the Narrow Class Group and the Negative Pell Equation.” Forum of Mathematics, Sigma, vol. 10, e46, Cambridge University Press, 2022, doi:10.1017/fms.2022.40.
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arXiv 1908.01752

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