Geodesic
Fundamental solutions to Pell equation with prescribed size
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 46-56 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove that the number of parameters $D$ up to a fixed $x\geq2$ such that the fundamental solution $\varepsilon_D$ to the Pell equation $T^2-DU^2=1$ lies between $D^{\frac12+\alpha_1}$ and $D^{\frac12+\alpha_2}$ is greater than $\sqrt x\log^2x$ up to a constant as long as $\alpha_1\alpha_2$ and $\alpha_13/2$. The starting point of the proof is a reduction step already used by the authors in earlier works. This approach is amenable to analytic methods. Along the same lines, and inspired by the work of Dirichlet, we show that the set of parameters $D\leq x$ for which $\log\varepsilon_D$ is larger than $D^\frac14$ has a cardinality essentially larger than $x^\frac14\log^2x$. 
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     author = {\'Etienne Fouvry and Florent Jouve},
     title = {Fundamental solutions to {Pell} equation with prescribed size},
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}
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Étienne Fouvry; Florent Jouve. Fundamental solutions to Pell equation with prescribed size. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and analysis, Tome 276 (2012), pp. 46-56. http://geodesic.mathdoc.fr/item/TM_2012_276_a4/

[1] Fouvry E., On the size of the fundamental solution of Pell equation, Preprint, Univ. Paris-Sud, 2010 | MR

[2] Fouvry E., Jouve F., A positive density of fundamental discriminants with large regulator, Preprint, Univ. Paris-Sud, 2011 | MR

[3] Fouvry E., Jouve F., Size of regulators and consecutive square-free numbers, Preprint, Univ. Paris-Sud, 2011 | MR

[4] Golubeva E.P., “Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields”, J. Sov. Math., 52:3 (1990), 3049–3056 | DOI | MR

[5] Hooley C., “On the Pellian equation and the class number of indefinite binary quadratic forms”, J. reine angew. Math., 353 (1984), 98–131 | MR | Zbl

[6] Lejeune Dirichlet G., “Sur une propriété des formes quadratiques à déterminant positif”, G. Lejeune Dirichlet's Werke, Bd. 2, Chelsea Publ. Co., Bronx, NY, 1969, 191–194

[7] Zagier D.B., Zetafunktionen und quadratische Körper: Eine Einführung in die höhere Zahlentheorie, Hochschultext, Springer, Berlin, 1981 | MR | Zbl