[seqfan] 72 is the only pronic and powerful?

Tanya Khovanova mathoflove-seqfan at yahoo.com
Fri Nov 30 17:30:35 CET 2012


Dear SeqFans,


I just received the following email from Paul Wright:

"Just thought I'd share a property that I didn't see listed on NumberGossip.com. 72 is the ONLY number that is both pronic and powerful, since 8 and 9 are the only consecutive non-trivial powers, as demonstrated by Mihăilescu's 2002 proof of Catalan's Conjecture.  NumberGossip lists 72 as both pronic and powerful, but doesn't mention that it is, in fact, the only number which satisfies both conditions."
There is a flaw in this reasoning. Suppose n and n+1 are both powerful, but not powers. Then their product will be powerful and pronic (pronic means the product of two consecutive numbers). I wonder if the proof of Catalan's conjecture extends to powerful numbers, not just powers.

Does anyone know anything about this?

Tanya


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