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

[Charles Greathouse]

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

Charles Greathouse
Analyst/Programmer
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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