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

[Emmanuel Vantieghem]

Dear Sir,

The following numbers are also missing in your list :
86149711981264908618 (smallest solution for the Pell equation with  y = 13)
5484296027914919579181500526692857773246 ( ... y = 37)
86561658417582391418287752764846216523628730400009366339782030 ( ... y =
29).

Sincerely yours

Emmanuel.
________________________

2012/11/16 Robert G. Wilson v <rgwv at rgwv.com>

> SeqFans,
>
>         Here is what I get so far.
>
>   682
>   1268860318
>   1459639851109444
>   2360712083917682
>   4392100110703410665318
>   8171493471761113423918890682
>   15203047261220215902863544865414318
>   28285239023397517753374058381589688919682
>   12439333951782387734360136352377558500557329868
>   52624630632537831937855708654927989510825107318
>   97908020042547086005693272723322840570529500826004682
>   182157675473066143788787784683258842527134067238269233744318
>   338903990902190706366709548741628016731324554988094903342010145682
>   630530197265361847138377315870912654087912613063713457410764922787325318
>
>
> 1060104169164423772274446059994682269961159864044458330444625185013342888064
> 84
>   ...
>
>
> 4279104078004899022080917697403989475430634859934233351021743853386879637099
>
> 3207437700222532927350542021925898943115140846846046603791372889127160448480
>
> 6456648567809059292918190921243953601798103132062280668429245895045322132474
>
> 8652845281679499196042278242733025358631331870513056058535101653128688454846
>
> 1633431386777413294481337916503669521364720759917056738645910768971999734826
> 6905616
>
> Sincerely yours, 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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