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

Robert G. Wilson v rgwv at rgwv.com
Sat Nov 17 21:34:12 CET 2012

```Dear Dave,

I shall fix that. I stopped my search for y = 1000.

Thank you,Bob.

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of David
Applegate
Sent: Saturday, November 17, 2012 2:53 PM
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3

>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:

5043617194272029913364566078547451446316984671863571394618629438567055488737
5103650385467064293443177111883953972061487227064158527982637862759949010755
4730442812986876349873204797074566710735715200459611061320331976123481729976
9789190056278682375591839756508391840892607461606282343190126491435438342581
8083314125039312356615694619357383418753405612306988812544177060367658367869
9793908998590140192296919980775332389070479327813125029691479149355047964478
0258749545731057099052225751216566136144215591768435540373383625312511829897
7436694130377409783079953451166544290906373051002958455620153233730808606863
1720378281200634363771767291235403441974199453301756891047489801679258681100
4199728246583492535760134916787230071315537845663096897549801990689118655222
8633763718730951124388978352259799746175613393161832870596287828081655596881
1866862402591417790061865838626350943356253902248543422248812377308274899432
4805367576771592746351839211407432321866139451372103319091229248470465073335
87
67186053966

from y=1241:

2934187435092306074934075674570536350861904476689894534678067350823392099612
0945537513774038371445547026219644810486028867939533850702160423499823928684
6356408022746965555198348426602973959396338341802633907851118228375077803771
3600182343966557401729459587177265555710454215465327273677110472032780376691
1263015328800759848173682805290581082699561868571998488512469845122829038969
9775823435917254672971554763634657191267701475079683804407759625101526579515
5127234282052129024372279819247630465145168476307613607798404833980452747380
8536730710706506183158657806055709775832

from 2581:

6303560856193916584805250679816201043130906409013330264385129129241303867731
5815119137414859643370000024344998868512932710661264241945406511095170346261
6935165253145753058429241477151157683135436573014282089259365428660704813319
1688965568683316013278144301487714015815456919258776890263162758249995173257
2264701588994712888445350670853335230730022211641153683750444416693516549642
3020884450857997313095079972910289085725289962945364360382056567710467709762
3188820868061397640424017870412510

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
> > >
> > 63053019726536184713837731587091265408791261306371345741076492278732
> > 53
> > 18
> > >
> > >
> > 10601041691644237722744460599946822699611598640444583304446251850133
> > 42
> > 888064
> > > 84
> > >   ...
> > >
> > >
> > 42791040780048990220809176974039894754306348599342333510217438533868
> > 79
> > 637099
> > >
> > 32074377002225329273505420219258989431151408468460466037913728891271
> > 60
> > 448480
> > >
> > 64566485678090592929181909212439536017981031320622806684292458950453
> > 22
> > 132474
> > >
> > 86528452816794991960422782427330253586313318705130560585351016531286
> > 88
> > 454846
> > >
> > 16334313867774132944813379165036695213647207599170567386459107689719
> > 99
> > 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
> Email: njasloane at gmail.com

> _______________________________________________

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

> _______________________________________________

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

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```