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On Some Exponential Equations of S.~S.~Pillai

Open Access article
 Printed: Oct 2001
  • Michael A. Bennett
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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 $a,b \geq 2$. 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.
MSC Classifications: 11D61, 11D45, 11J86 show english descriptions Exponential equations
Counting solutions of Diophantine equations
Linear forms in logarithms; Baker's method
11D61 - Exponential equations
11D45 - Counting solutions of Diophantine equations
11J86 - Linear forms in logarithms; Baker's method

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