A puzzle – higher powers
July 29, 2007 pm31 8:30 pm
Eight and nine are the third power of two and the second power of three, respectively. Are there any other examples of consecutive numbers which are of the form 
Hm. Maybe this is not such a quick one. And it really is a question, rather than a puzzle. I don’t know the answer.
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JD2718 
This is a hard problem. I know it has appeared on the forums at http://www.artofproblemsolving.com, but I wasn’t actually able to find a thread on which it was solved and I don’t know the proof. It is in fact true that 8 and 9 are the only pair of consecutive perfect powers.
Ah, even harder than I thought: this result was published as a conjecture of Catalan in 1844; it was not proven until very recently, by Preda Mihailescu, using some powerful technology. There’s an article about it here: http://www.ams.org/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf
This reminds me of Merisenne (sp?) primes. 2^n being even and 2^n-1 being odd… this question isn’t the converse of Merisenne primes, but seems to fit in the opposite camp and is related.
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