Differences between Perfect Powers 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. Categories:11D61, 11D45