[seqfan] Re: 1^5+2^5+...+n^5 is a square

Max Alekseyev maxale at gmail.com
Mon Oct 22 18:18:22 CEST 2012


This reduces to generalized Pell equation:
6q^2 = (2n+1)^2 - 3
where q = 2m/(n*(n+1)).
So there is no surprise that solutions satisfy linear recurrent equation.
Max

On Mon, Oct 22, 2012 at 10:23 AM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> It was recently asked (on MathOverflow, I think) whether the formula on A031138:
>
> a(n) =11*(a(n-1)-a(n-2)) + a(n-3)
>
> was proved or merely conjectural. Of course it should be proved to be
> included as it is, but would someone verify this?
>
> This is of course a 6th-degree Diophantine equation:
>
> 12m^2 = n^2 (n+1)^2 (2n^2 + 2n - 1)
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
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>
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