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Differences between Perfect Powers |
Canad. Math. Bull. 51(2008), 337-347
Published:2008-09-01
Printed: Sep 2008
Printed: Sep 2008
- Michael A. Bennett
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 |