Finiteness of integral values for the ratio of two linear recurrences

被引:25
作者
Corvaja, P
Zannier, U
机构
[1] Univ Udine, Dipartimento Matemat & Informat, I-33100 Udine, Italy
[2] Inf, I-33100 Udine, Italy
[3] DCA, IUAV, I-30135 Venice, Italy
关键词
D O I
10.1007/s002220200221
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let {F(n)}(nis an element ofN), {G(n)}(nis an element ofN), be linear recurrent sequences. In this paper we are concerned with the well-known diophantine problem of the finiteness of the set X 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 X coincides with N, then {F(n)/G(n)}(nis an element ofN) is itself a linear recurrence sequence. Here we shall prove that if X 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 X in the cases when the polynomial P cannot be taken a constant.
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收藏
页码:431 / 451
页数:21
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