Some arithmetic properties of recurrence sequences
type: Item
instance of: scholarly article
title: Some arithmetic properties of recurrence sequences
zbMATH Open document ID: 0714.11009
DOI: 10.1007/BF01170895
author: Igor E. Shparlinski
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: Let \(\{u(x)\}^{\infty}_{x=1}\) be a linear recursive sequence of integers. This sequence is called nondegenerate if none of the roots and none of the quotients of the roots of its characteristic polynomial are roots of unity. Let S be a set of r prime numbers. For integers \(0<M,N\) let A(S;M,N) denote the number of integers \(M\leq t\leq M+N\) such that u(t) is divisible only by elements from S. It is proved that if \(\{u(x)\}^{\infty}_{x=1}\) is nondegenerate then \(A(S;M,N)\ll r(N \ln (M+N)/M+1),\) and \[ (1/N)\sum^{N}_{u(x)\neq 0;x=1}\phi (| u(x)|)/| u(x)| =\gamma +O(\frac{\ln \ln N}{\ln N}), \] where \(\gamma >0\) is a constant and \(\phi\) denotes Euler's function.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 4175011
zbMATH Keywords: divisibility
zbMATH Keywords: linear recurrence sequences
reviewed by: Attila Pethő
MaRDI profile type: Publication
cites work: On the growth of recurrence sequences
cites work: Q3338318
cites work: Q5638376
cites work: Q3773941
cites work: Q3207067
description: scientific article; zbMATH DE number 4175011