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Effective measures of irrationality for certain algebraic numbers

Part of: Diophantine approximation, transcendental number theory Computational number theory Diophantine equations

Published online by Cambridge University Press:  09 April 2009

Michael A. Bennett
Michael A. Bennett
Affiliation:
Department of Mathematics University of MichiganAnn Arbor, MI 48109USA e-mail: [email protected]
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Abstract

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In this paper, we derive a number of explicit lower bounds for rational approximation to certain cubic irrationalities, proving, for example, that for any non-zero integers p and q. A number of these irrationality measures improve known results, including those for . Some Diophantine consequences are briefly discussed.

Keywords

Effective irrationality measurescubic irrationalitiesPadé approximantssimultaneous figurate numbers

MSC classification

Secondary: 11J68: Approximation to algebraic numbers 11J82: Measures of irrationality and of transcendence 11D25: Cubic and quartic equations 11Y50: Computer solution of Diophantine equations 11Y65: Continued fraction calculations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 62 , Issue 3 , June 1997 , pp. 329 - 344
Copyright
Copyright © Australian Mathematical Society 1997

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