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Effective irrationality measures for real and p-adic roots of rational numbers close to 1, with an application to parametric families of Thue–Mahler equations

Published online by Cambridge University Press:  27 September 2016

YANN BUGEAUD
YANN BUGEAUD*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7, rue René Descartes, 67084 Strasbourg, France. e-mail: yann.bugeaud@math.unistra.fr
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Abstract

We show how the theory of linear forms in two logarithms allows one to get very good effective irrationality measures for nth roots of rational numbers a/b, when a is very close to b. We give a p-adic analogue of this result under the assumption that a is p-adically very close to b, that is, that a large power of p divides a−b. As an application, we solve completely certain families of Thue–Mahler equations. Our results illustrate, admittedly in a very special situation, the strength of the known estimates for linear forms in logarithms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 99 - 108
Copyright
Copyright © Cambridge Philosophical Society 2016 

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