Sums of \(S\)-units in recurrence sequences
type: Item
instance of: scholarly article
title: Sums of \(S\)-units in recurrence sequences
zbMATH Open document ID: 1465.11084
published in: Journal of Number Theory
publication date: 2018-12-03T00:00:00Z
review text: Let \( S=\{p_1, \ldots, p_s\} \) be a set of primes, and write \( \mathbb{Z}_S \) for the set of those integers which have no prime divisors outside \( S \). Let \( U_n \) be a linear recurrence sequence of integers. A linear recurrence sequence of order \(r\) is the sequence \( (U_n)_{n\ge 0} \) satisfying the relation \begin{align*} U_n=a_1U_{n-1}+\cdots+a_rU_{n-r}, \end{align*} where \( a_1, \ldots, a_r \) are integers with \( a_r\ne 0 \) and \( U_0, \ldots, U_{r-1} \) are integrs not all zero. The characteristic polynomial of \( U_n \) is defined by \begin{align*} f(x):=x^r-a_1x^{r-1}-\cdots-a_r=\prod_{i=1}^{t}(x-\alpha_i)^{m_i}, \end{align*} where \( \alpha_1, \ldots, \alpha_t \) are distinct algebraic numbers and \( m_1, \ldots, m_t \) are integers. The sequence \( (U_n)_{n\ge 0} \) is called \textit{degenerate} if there are integers \( i, j \) with \( 1\le i<j\le t \) such that \( \alpha_i/\alpha_j \) is not a root of unity; otherwise it is called \textit{non-degenerate}. If for some \( i \) with \( 1\le i\le t \) we have \( |\alpha_i|>|\alpha_j| \) for all \( j \) with \( 1\le j\le t \) and \( j\ne i \), then \( \alpha_i \) is called a \textit{dominant root} of the sequence \( (U_n)_{n\ge 0} \). In the paper under review, the authors study the Diophantine equation \[U_n=w_1+\cdots+w_k \tag{1}\] with some arbitrary but fixed \( k\ge 1 \), in unknown \( n\ge 0 \) and \( w_1, \ldots, w_k \in \mathbb{Z}_S \). Their major aim is to establish finiteness results for the solutions of (1). Their main results are the following. Theorem 2. Let \(U_n\) be a non-degenerate recurrence sequence of order \(r\), and write \(f (x)\) for the characteristic polynomial of \(U_n\). Suppose that at least one of the following two conditions is valid: \begin{itemize} \item[(i)]\( f(x) \) has at least one irrational root, \item[(ii)]\( f(0) \) has a prime divisor outside \( S \). \end{itemize} Then for any fixed \(k \ge 1\), equation (1) is solvable at most for finitely many \(n\). F…
reviewed by: Mahadi Ddamulira
Mathematics Subject Classification ID: 11D61
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 6987939
zbMATH Keywords: recurrence sequences
zbMATH Keywords: sums of \(S\)-units
zbMATH Keywords: \(S\)-unit equations
zbMATH Keywords: polynomial-exponential equations
zbMATH Keywords: Baker's method
author: Attila Bérczes
author: Lajos Hajdu
author: István Pink
author: S. S. Rout
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.10.009
OpenAlex ID: W2901369171
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Wikidata QID: Q128943260
DOI: 10.1016/J.JNT.2018.10.009
description: scientific article; zbMATH DE number 6987939