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I'd be curious to know how often the closest pair has $\min(m,n)>2$ (or even $>1$). Bennett shows that $|a^m-b^n|<\frac{\max(\sqrt{a^m},\sqrt{b^n})}{4}$ happens at most once. However this might not happen for most pairs $a,b$ and even when this happens, it might not be a min. (And it seems rare for it to happen with $\min(m,n)>1$.) A rather spectacular (to my mind) case is $13^3-3^7=10$. The first few convergents of the continued fraction for the ratio of the logs are $1/2,2/5,3/7,239/558$ This shows that 3/7 is an extremely good approximation (of course). But the closest pair is $13^1-3^2=4$.