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 N 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 N coincides with N, then {F(n)/G(n)}_(n∈N) is itself a linear recurrence sequence. Here we shall prove that if N 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 N in the cases when the polynomial P cannot be taken a constant.
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