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PERFECT POWERS WITH THREE DIGITS

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

Published online by Cambridge University Press:  06 August 2013

Michael A. Bennett  and
Yann Bugeaud
Michael A. Bennett
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, V6T 1Z2,Canada email bennett@math.ubc.ca
Yann Bugeaud
Affiliation:
Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg,France email bugeaud@math.unistra.fr
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Abstract

We solve the equation ${x}^{a} + {x}^{b} + 1= {y}^{q} $ in positive integers $x, y, a, b$ and $q$ with $a\gt b$ and $q\geq 2$ coprime to $\phi (x)$. This requires a combination of a variety of techniques from effective Diophantine approximation, including lower bounds for linear forms in complex and $p$-adic logarithms, the hypergeometric method of Thue and Siegel applied $p$-adically, local methods, and the algorithmic resolution of Thue equations.

MSC classification

Secondary: 11A63: Radix representation; digital problems 11D61: Exponential equations 11J86: Linear forms in logarithms; Baker's method

Information

Type
Research Article
Information
Mathematika , Volume 60 , Issue 1 , January 2014 , pp. 66 - 84
Copyright
Copyright © University College London 2013 

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