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Size of regulators and consecutive square–free numbers

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Abstract

We prove that, for almost all D, the fundamental solution \({x_{0}+y_{0}\sqrt D}\) of the associated Pell equation x 2−Dy 2 = 1 is greater than D 1.749···. We also show a strong link between this question and the error term in the asymptotic formula of the number of pairs of consecutive square-free numbers.

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  1. Laboratoire de Mathématique, Université Paris–Sud, CNRS (UMR 8628), Orsay, 91405, France

    Étienne Fouvry & Florent Jouve

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  1. Étienne Fouvry
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  2. Florent Jouve
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Correspondence to Étienne Fouvry.

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Fouvry, É., Jouve, F. Size of regulators and consecutive square–free numbers. Math. Z. 273, 869–882 (2013). https://doi.org/10.1007/s00209-012-1035-7

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  • DOI: https://doi.org/10.1007/s00209-012-1035-7

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