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Differences between Perfect Powers

Published:2008-09-01
Printed: Sep 2008
• Michael A. Bennett
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Abstract

We apply the hypergeometric method of Thue and Siegel to prove that if $a$ and $b$ are positive integers, then the inequality $0 <| a^x - b^y | < \frac{1}{4} \, \max \{ a^{x/2}, b^{y/2} \}$ has at most a single solution in positive integers $x$ and $y$. This essentially sharpens a classic result of LeVeque.
 MSC Classifications: 11D61 - Exponential equations 11D45 - Counting solutions of Diophantine equations