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Differences between Perfect Powers

Published online by Cambridge University Press:  20 November 2018

Michael A. Bennett
Michael A. Bennett*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2. e-mail: [email protected]
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Abstract

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We apply the hypergeometric method of Thue and Siegel to prove that if $a$ and $b$ are positive integers, then the inequality $0\,<\,\left| {{a}^{x}}\,-\,{{b}^{y}} \right|\,<\,\frac{1}{4}\,\max \{{{a}^{x/2}},\,{{b}^{y/2}}\}$ has at most a single solution in positive integers $x$ and $y$. This essentially sharpens a classic result of LeVeque.

Keywords

11D6111D45
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 3 , 01 September 2008 , pp. 337 - 347
Copyright
Copyright © Canadian Mathematical Society 2008

References

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