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

[Robert G. Wilson v]

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

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