# Finiteness of integral values for the ratio of two linear recurrences

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## 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## Preview

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