Finiteness of integral values for the ratio of two linear recurrences
type: Item
instance of: scholarly article
title: Finiteness of integral values for the ratio of two linear recurrences
zbMATH Open document ID: 1026.11021
author: Pietro Corvaja
author: Umberto Zannier
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: Some years ago, the reviewer showed [C. R. Acad. Sci., Paris, Sér. I 306, 97-102 (1998; Zbl 0635.10007); see also \textit{R. Rumely}, Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 349-382, 383-409 (1988; Zbl 0661.10017)] that if every term of the term-by-term quotient of two linear recurrence sequences is an integer then that `Hadamard quotient' must itself be a linear recurrence sequence. Here the authors prove a best possible such result, thus with the minimal hypothesis that just infinitely many of those quotients be integral. The core idea of the argument is quite different from that of the reviewer. The \(p\)-adic subspace theorem of Schmidt and Schlickewei is applied to the difference of the quotient and an appropriate approximation thereto. That has two advantages. First, one needs the integrality data only for infinitely many terms rather than for all terms and, second, the argument in the general case is not of a materially different nature from that of the `dominant root' case. The authors give a very clear explanation of their powerful arguments and of the manifold implications of their result. There is also useful discussion of related quantitative considerations.
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 1965430
zbMATH Keywords: linear recurrence sequences
zbMATH Keywords: Hadamard quotient
zbMATH Keywords: \(p\)-adic subspace theorem
zbMATH Keywords: integrality
MaRDI profile type: Publication
full work available at URL: s002220200221
OpenAlex ID: W2094789907
Wikidata QID: Q56699707
DOI: 10.1007/S002220200221
description: scientific article; zbMATH DE number 1965430