# [seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

Charles Greathouse charles.greathouse at case.edu
Fri Nov 16 19:11:32 CET 2012

```It could go one way or the other. 0 meets the definition of a powerful
number, too, but it's not in A001694.

Lacking reason to do otherwise I'd stick to positive numbers. We're clearly
not using integers, since then the sequence would never start.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Fri, Nov 16, 2012 at 1:06 PM, Robert G. Wilson v <rgwv at rgwv.com> wrote:

> SeqFans,
>
> Shouldn't 0 also be in the sequence since it satisfies the criteria?
>
> Bob.
>
> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Charles
> Greathouse
> Sent: Friday, November 16, 2012 11:24 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3
>
> Good catch, Georgi.
>
> I wouldn't add a new sequence, just correct this one. (Unless someone feels
> that this would be worthwhile?)
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Fri, Nov 16, 2012 at 10:56 AM, Georgi Guninski
> <guninski at guninski.com>wrote:
>
> > A175155 Numbers n satisfying n^2 + 1 = x^2 y^3
> >
> > I am not sure this is entirely correct:
> > >This sequence is infinite. The fundamental solution of n^2 + 1 = x^2
> > >y^3
> > is (n,x,y) = (682,61,5), that mean the Pellian equation n^2 - 125x^2 =
> > -1 has the solution (n,x) = (682,61) =(n(1),x(1)). Then, this Pellian
> > equation admit an infinity solutions (n(2k+1),x(2k+1))
> >
> > This indeed is a family of solutions giving the smallest one, but
> > there are infinitely many other solutions arising from
> > x^2 - k^3 y^2 = -1
> >
> > In particular n=1459639851109444 is missing from the sequence.
> > n^2 + 1 = 17^3 * 79153^2 * 263090369^2
> >
> > I suggest:
> > 1. Adding the missing term and other low hanging fruit from pell eqs
> > 2. Indicating that terms might be missing (the sequence contains a
> > 22 digit number and I suppose it is infeasible to find all terms up to
> > it)
> >
> > Should I submit another sequence that currently numerically coincides
> > with A175155?
> >
> > Solution x to x^2 - 125 y^2 = -1
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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```