[seqfan] Re: 72 is the only pronic and powerful?

Charles Greathouse charles.greathouse at case.edu
Fri Nov 30 17:48:45 CET 2012

The next few powerful pronic numbers are 83232, 456300, 96049800,
148048056, 55330565400, 110841386112, ... and there are infinitely many.

Charles Greathouse
Case Western Reserve University

On Fri, Nov 30, 2012 at 11:30 AM, Tanya Khovanova <
mathoflove-seqfan at yahoo.com> wrote:

> 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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