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


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.


Natural Number Arithmetic Progression Linear Recurrence Recurrence Sequence Linear Recurrence Sequence 
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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:
  2. 2.I.U.A.V. – DCA, S. Croce, 191, 30135 Venezia, Italy (e-mail:

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