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$S$-PARTS OF TERMS OF INTEGER LINEAR RECURRENCE SEQUENCES

Part of: Sequences and sets Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  29 November 2017

Yann Bugeaud  and
Jan-Hendrik Evertse
Yann Bugeaud
Affiliation:
Université de Strasbourg, CNRS, IRMA UMR 7501, 7, rue René Descartes, 67000 Strasbourg, France email [email protected]
Jan-Hendrik Evertse
Affiliation:
Universiteit Leiden, Mathematisch Instituut, Postbus 9512, 2300 RA Leiden, The Netherlands email [email protected]
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Abstract

Let $S=\{q_{1},\ldots ,q_{s}\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}M$, where $r_{1},\ldots ,r_{s}$ are non-negative integers and $M$ is an integer relatively prime to $q_{1}\cdots q_{s}$. We define the $S$-part $[m]_{S}$ of $m$ by $[m]_{S}:=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}$. Let $(u_{n})_{n\geqslant 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\unicode[STIX]{x1D700}>0$, there exists an integer $n_{0}$ such that $[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$ holds for $n>n_{0}$. Our proof is ineffective in the sense that it does not give an explicit value for $n_{0}$. Under various assumptions on $(u_{n})_{n\geqslant 0}$, we also give effective, but weaker, upper bounds for $[u_{n}]_{S}$ of the form $|u_{n}|^{1-c}$, where $c$ is positive and depends only on $(u_{n})_{n\geqslant 0}$ and $S$.

MSC classification

Primary: 11B37: Recurrences 11J86: Linear forms in logarithms; Baker's method 11J87: Schmidt Subspace Theorem and applications
Type
Research Article
Information
Mathematika , Volume 63 , Issue 3 , 2017 , pp. 840 - 851
Copyright
Copyright © University College London 2017 

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