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Mathematische Zeitschrift

, Volume 273, Issue 3–4, pp 869–882 | Cite as

Size of regulators and consecutive square–free numbers

  • Étienne FouvryEmail author
  • Florent Jouve
Article

Abstract

We prove that, for almost all D, the fundamental solution \({x_{0}+y_{0}\sqrt D}\) of the associated Pell equation x 2Dy 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.

Mathematics Subject Classification

Primary 11D09 Secondary 11L05 

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References

  1. 1.
    Carlitz L.: On a problem in additive arithmetic. Quart. J. Math. Oxford Ser. 3, 273–290 (1932)zbMATHCrossRefGoogle Scholar
  2. 2.
    Cremona J.E., Odoni R.W.K.: Some density results for negative Pell equations; an application of graph theory. J. London Math. Soc. 39(1), 16–28 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Deligne, P.: Cohomologie étale. Séminaire de Géométrie Algébrique du Bois-Marie SGA 4\({\frac{1}{2}}\). Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. Lecture Notes in Math, vol. 569. Springer-Verlag, Berlin-New York (1977)Google Scholar
  4. 4.
    Fouvry, E.: On the size of the fundamental solution of Pell equation (2011, submitted) http://www.math.u-psud.fr/~fouvry
  5. 5.
    Fouvry, E., Jouve, F.: A positive density of fundamental discriminants with large regulator. (2011, submitted) http://www.math.u-psud.fr/~fouvry
  6. 6.
    Fouvry E., Jouve F.: Fundamental solutions to Pell equation with prescribed size. Proc. Steklov Institute Math. 279, 40–50 (2012)CrossRefGoogle Scholar
  7. 7.
    Friedlander J., Iwaniec H.: Opera de Cribro, Colloquium Publications, 57. American Mathematical Society, Providence (2010)Google Scholar
  8. 8.
    Heath-Brown D.R.: The square sieve and consecutive square–free numbers. Math. Ann. 266, 251–284 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hooley C.: On the Pellian equation and the class number of indefinite binary quadratic forms. J. für Reine Angew Math. 353, 98–131 (1984)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Legendre, A.M.: Théorie des Nombres, Tome 1, Quatrième Edition, Librairie A. Blanchard, Paris (1955)Google Scholar
  11. 11.
    Lejeune–Dirichlet, G.: Einige neue Sätze über unbestimmte Gleichungen, G. Lejeune Dirichlet’s Werke, vol. 1 & 2, pp. 219–236. Chelsea Publishing Company Bronx, New York, Erster Band (1969)Google Scholar
  12. 12.
    Lemmermeyer, F.: Higher descent on Pell conics. I. ArXiv:math/0311309v1, [math.NT] 18 nov (2003)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Laboratoire de MathématiqueUniversité Paris–Sud, CNRS (UMR 8628)OrsayFrance

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