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Inventiones mathematicae

, Volume 149, Issue 2, pp 431–451 | Cite as

Finiteness of integral values for the ratio of two linear recurrences

  • Pietro Corvaja
  • Umberto Zannier

Abstract.

Let {F(n)} n N ,{G(n)} n N , be linear recurrent sequences. In this paper we are concerned with the well-known diophantine problem of the finiteness of the set ? of natural numbers n such that F(n)/G(n) is an integer. In this direction we have for instance a deep theorem of van der Poorten; solving a conjecture of Pisot, he established that if ? coincides with N, then {F(n)/G(n)} n N is itself a linear recurrence sequence. Here we shall prove that if ? is an infinite set, then there exists a nonzero polynomial P such that P(n)F(n)/G(n) coincides with a linear recurrence for all n in a suitable arithmetic progression. Examples like F(n)=2 n -2, G(n)=n+2 n +(-2) n , show that our conclusion is in a sense best-possible. In the proofs we introduce a new method to cope with a notorious crucial difficulty related to the existence of a so-called dominant root. In an appendix we shall also prove a zero-density result for ? in the cases when the polynomial P cannot be taken a constant.

Keywords

Natural Number Arithmetic Progression Linear Recurrence Recurrence Sequence Linear Recurrence Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Pietro Corvaja
    • 1
  • Umberto Zannier
    • 2
  1. 1.Dipartimento di Mat. e Inf., via delle Scienze, 206, 33100 Udine, Italy (e-mail: corvaja@dimi.uniud.it)IT
  2. 2.I.U.A.V. – DCA, S. Croce, 191, 30135 Venezia, Italy (e-mail: zannier@iuav.it)IT

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