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

[David Applegate]

>From pounding with Maple, I found 3 terms missing from the b-file.
One (from y=37) occurs just before the end of the b-file, and contains
1001 digits, so perhaps the right answer is that the current b-file
shouldn't contain the 1001-digit entry.  The other two are from larger
y (1241 and 2581) which give small-ish solutions.

from y=37:

504361719427202991336456607854745144631698467186357139461862943856705548873751036503854670642934431771118839539720614872270641585279826378627599490107554730442812986876349873204797074566710735715200459611061320331976123481729976978919005627868237559183975650839184089260746160628234319012649143543834258180833141250393123566156946193573834187534056123069888125441770603676583678699793908998590140192296919980775332389070479327813125029691479149355047964478025874954573105709905222575121656613614421559176843554037338362531251182989774366941303774097830799534511665442909063730510029584556201532337308086068631720378281200634363771767291235403441974199453301756891047489801679258681100419972824658349253576013491678723007131553784566309689754980199068911865522286337637187309511243889783522597997461756133931618328705962878280816555968811866862402591417790061865838626350943356253902248543422248812377308274899432480536757677159274635183921140743232186613945137210331909122924847046507333587
 67186053966

from y=1241:

29341874350923060749340756745705363508619044766898945346780673508233920996120945537513774038371445547026219644810486028867939533850702160423499823928684635640802274696555519834842660297395939633834180263390785111822837507780377136001823439665574017294595871772655557104542154653272736771104720327803766911263015328800759848173682805290581082699561868571998488512469845122829038969977582343591725467297155476363465719126770147507968380440775962510152657951551272342820521290243722798192476304651451684763076136077984048339804527473808536730710706506183158657806055709775832

from 2581:

6303560856193916584805250679816201043130906409013330264385129129241303867731581511913741485964337000002434499886851293271066126424194540651109517034626169351652531457530584292414771511576831354365730142820892593654286607048133191688965568683316013278144301487714015815456919258776890263162758249995173257226470158899471288844535067085333523073002221164115368375044441669351654964230208844508579973130950799729102890857252899629453643603820565677104677097623188820868061397640424017870412510

As long as Maple is correctly solving Pell equations, these are the
only solutions for y <= 3000 missing from the b file.  But as far as I
know there could still be smaller solutions from larger y.

-Dave

> From seqfan-bounces at list.seqfan.eu Sat Nov 17 14:09:06 2012
> From: "Robert G. Wilson v" <rgwv at rgwv.com>
> To: "'Sequence Fanatics Discussion list'" <seqfan at list.seqfan.eu>
> Date: Sat, 17 Nov 2012 14:08:47 -0500
> Subject: [seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

> Consider it done. Bob.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil Sloane
> Sent: Saturday, November 17, 2012 8:05 AM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

> I agree that 0 should be in the sequence. I added it and added the condition
> that x and y should be positive. The b-file is now wrong, since 0 is
> missing, but I assume Bob Wilson will be changing it anyway when he responds
> to Georgi's comment.
> Neil

> On Sat, Nov 17, 2012 at 1:59 AM, Georgi Guninski
> <guninski at guninski.com>wrote:

> > Robert,
> >
> > You submitted a b-file. Are you sure you are not missing terms?
> >
> > Is it possible for large k
> > x^2 - k^3 y^2 = -1
> > to have relatively small solution x?
> > (Probably this depends on the length of the period of the c.f.).
> >
> >
> >
> > On Fri, Nov 16, 2012 at 12:54:17PM -0500, Robert G. Wilson v wrote:
> > > SeqFans,
> > >
> > >       Here is what I get so far.
> > >
> > >   682
> > >   1268860318
> > >   1459639851109444
> > >   2360712083917682
> > >   4392100110703410665318
> > >   8171493471761113423918890682
> > >   15203047261220215902863544865414318
> > >   28285239023397517753374058381589688919682
> > >   12439333951782387734360136352377558500557329868
> > >   52624630632537831937855708654927989510825107318
> > >   97908020042547086005693272723322840570529500826004682
> > >   182157675473066143788787784683258842527134067238269233744318
> > >   338903990902190706366709548741628016731324554988094903342010145682
> > >
> > 6305301972653618471383773158709126540879126130637134574107649227873253
> > 18
> > >
> > >
> > 1060104169164423772274446059994682269961159864044458330444625185013342
> > 888064
> > > 84
> > >   ...
> > >
> > >
> > 4279104078004899022080917697403989475430634859934233351021743853386879
> > 637099
> > >
> > 3207437700222532927350542021925898943115140846846046603791372889127160
> > 448480
> > >
> > 6456648567809059292918190921243953601798103132062280668429245895045322
> > 132474
> > >
> > 8652845281679499196042278242733025358631331870513056058535101653128688
> > 454846
> > >
> > 1633431386777413294481337916503669521364720759917056738645910768971999
> > 734826
> > > 6905616
> > >
> > > Sincerely yours, Bob.
> > >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >

> --
> Dear Friends, I have now retired from AT&T. New coordinates:

> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com

> _______________________________________________

> Seqfan Mailing list - http://list.seqfan.eu/

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