Powers of 2 and 3
Exploring power sequences can be a lot of fun! Many famous problems have arisen from playing with such sequences; for example, Fermat's Last Theorem (proved three and a half centuries after it was first stated by Fermat), Catalan's problem ("The only two perfect powers that differ by 1 are 8 and 9"; see http://mathworld.wolfram.com/CatalansDiophantineProblem.html; the theorem was proved very recently, in 2002, by the Swiss mathematician Preda Mihăilescu), and Beal's problem (which is still open).
On my website www.mathcelebration.com, which I launched more than a year back, and which I use to upload articles I have written for teachers and students, I have an article all about the powers of 2 and 3. Here are two links to it: Do have a look at it!
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