Article ID Journal Published Year Pages File Type
5772604 Journal of Number Theory 2017 13 Pages PDF
Abstract
Let F and G be linear recurrences over a number field K, and let R be a finitely generated subring of K. Furthermore, let N be the set of positive integers n such that G(n)≠0 and F(n)/G(n)∈R. Under mild hypothesis, Corvaja and Zannier proved that N has zero asymptotic density. We prove that #(N∩[1,x])≪x⋅(log⁡log⁡x/log⁡x)h for all x≥3, where h is a positive integer that can be computed in terms of F and G. Assuming the Hardy-Littlewood k-tuple conjecture, our result is optimal except for the term log⁡log⁡x.
Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory
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