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A Note on the Negative Pell Equation

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From Arithmetic to Zeta-Functions

Abstract

Nagell conjectured in the 1930s that the set of discriminants for which the negative Pell equation has an integral solution has an explicitly given positive proportion within the set of discriminants having no prime factor congruent to 3 modulo 4. In a series of papers, Fouvry and Klüners succeeded in showing that the order of magnitude of such discriminants up to x is indeed x(logx)−1∕2. Here we present a short independent argument that the order of magnitude is at least x(logx)−0. 62.

To the memory of Wolfgang Schwarz

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References

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Authors and Affiliations

  1. Universität Göttingen, Bunsenstr. 3-5, 37073, Göttingen, Germany

    Valentin Blomer

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  1. Valentin Blomer
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Correspondence to Valentin Blomer .

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Editors and Affiliations

  1. Institut für Mathematik und Angewandte Informatik, Stiftung Universität Hildesheim, Hildesheim, Germany

    Jürgen Sander

  2. Institut für Mathematik, Julius-Maximilians Universität Würzburg, Würzburg, Germany

    Jörn Steuding

  3. Institut für Mathematik, Julius-Maximilians Universität Würzburg, Würzburg, Germany

    Rasa Steuding

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Blomer, V. (2016). A Note on the Negative Pell Equation. In: Sander, J., Steuding, J., Steuding, R. (eds) From Arithmetic to Zeta-Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-28203-9_2

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