Some divisibilities amongst the terms of linear recurrences
type: Item
instance of: scholarly article
title: Some divisibilities amongst the terms of linear recurrences
zbMATH Open document ID: 1127.11012
DOI: 10.1007/BF02960862
author: Florian Luca
author: Igor E. Shparlinski
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2007-08-29T00:00:00Z
review text: Consider an integer linear recurrence sequence \((u_n)_{n \geq 0}\) of order \(r \geq 2\): \[ u_{n + r} = a_1u_{n + r - 1} + \dots + a_{r - 1}u_{n + 1} + a_ru_n \quad (n \geq 0), \] and let \(\psi(x)\) be its characteristic polynomial. The sequence \((u_n)_{n \geq 0}\) is called \textit{non-degenerate}, if \(\lambda^m \neq \mu^m\) for any \(m \in \mathbb N\) and any two distinct roots \(\lambda, \mu\) of \(\psi(x)\). Given an arithmetic function \(f\), let \(N_f(x)\) denote the number of integers \(n\), with \(n \leq x\), such that \(f(n) \mid u_n\). In this paper, the authors obtain bounds of the form \(N_f(x) = o(x)\) when \(f(n)\) is one of the functions \(\tau(n), \omega(n)\), or \(\Omega(n)\) (these are, respectively, the number of positive divisors of \(n\), the number of distinct prime divisors of \(n\), and the total number of prime divisors of \(n\)). Suppose that the sequence \((u_n)_{n \geq 0}\) is non-degenerate. When \(f(n) = \omega(n)\) or \(f(n) = \Omega(n)\), the authors prove that \(N_f(x) \ll x(\log\log\log x)^{-c_r}\), where \(c_r > 0\) is a constant described explicitly in the paper. They also show that when \(a_r\) is odd, one has \(N_{\tau}(x) \ll x(\log x)^{-2^{-r}}\). The authors note that the condition on \(a_r\) in the latter result cannot be entirely dispensed with.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 5184535
zbMATH Keywords: linear recurrence sequences
zbMATH Keywords: divisor function
zbMATH Keywords: number of prime divisors
reviewed by: Angel V. Kumchev
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02960862
OpenAlex ID: W1986759004
description: scientific article; zbMATH DE number 5184535