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Perfect powers with few binary digits and related Diophantine problems, II

Published online by Cambridge University Press:  16 August 2012

MICHAEL A. BENNETT ,
YANN BUGEAUD  and
MAURICE MIGNOTTE
MICHAEL A. BENNETT
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C. Canada V6T 1Z6. e-mail: [email protected]
YANN BUGEAUD
Affiliation:
Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France. e-mail: [email protected], [email protected]
MAURICE MIGNOTTE
Affiliation:
Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France. e-mail: [email protected], [email protected]
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Abstract

We prove that if q ≥ 5 is an integer, then every qth power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms with various local arguments.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 153 , Issue 3 , November 2012 , pp. 525 - 540
Copyright
Copyright © Cambridge Philosophical Society 2012

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