[seqfan] Re: Fixing A175155 Numbers n satisfying n^2 + 1 = x^2 y^3
Charles Greathouse
charles.greathouse at case.edu
Fri Nov 16 17:23:46 CET 2012
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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