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PERFECT POWERS WITH THREE DIGITS
Published online by Cambridge University Press: 06 August 2013
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
- Type
- Research Article
- Information
- Copyright
- Copyright © University College London 2013
References
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