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On Some Exponential Equations of S. S. Pillai

Published online by Cambridge University Press:  20 November 2018

Michael A. Bennett
Michael A. Bennett*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, IL 61801, USA. email: mabennet@math.uiuc.edu
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Abstract

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In this paper, we establish a number of theorems on the classic Diophantine equation of S. S. Pillai, ${{a}^{x}}-{{b}^{y}}=c$ , where $a,\,b$ and $c$ are given nonzero integers with $a,\,b\,\ge \,2$. In particular, we obtain the sharp result that there are at most two solutions in positive integers $x$ and $y$ and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Keywords

11D6111D4511J86
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 5 , 01 October 2001 , pp. 897 - 922
Copyright
Copyright © Canadian Mathematical Society 2001

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