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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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