[seqfan] Re: p = (k + 1)^2 - k = (m + 1)^3 - m and p = (k + 1)^2 + k = (m + 1)^2 + m

[Max Alekseyev]

Both
(k + 1)^2 - k = (m + 1)^3 - m
and
(k + 1)^2 + k = (m + 1)^3 + m
define elliptic curves with a finite number of integral points.
The first one has those for m = -2, -1, 0, 1, and 5, while the second
has m = -1, 0, and 2.
There are no others.
Max

On Fri, Sep 20, 2013 at 1:53 PM, юрий герасимов <2stepan at rambler.ru> wrote:
>
> Dear SeqFans,
> p = (k + 1)^2 - k = (m + 1)^3 - m:
>
> 7 = (2 + 1)^2 - 2 = (1 + 1)^3 - 1,
>
> 211 = (14 + 1)^2 -14 = (5 + 1)^3 - 5,
> What is the next one?
>
> p = (k + 1)^2 + k = (m + 1)^3 + m:
> 29 = (4 + 1)^2 + 4 = (2 + 1)^3 + 2,
> What is the next one?
> Best regards,
> JSG
>
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