Re: Primes again [really Mordell's equation]
- --- In email@example.com,
"mikeoakes2" <mikeoakes2@...> wrote:
> what led you to discover/investigate that particularIt's a record-holder for rank 5:
> rank-5 elliptic curve
> x^3 = y^2 + 28279
J Gebel, A. Petho, H.G. Zimmer,
On Mordell's Equation,
Compositio Mathematica 110: 335-367, 1998.
Then it was easy to use Magma, on line, to determine that
this curve has /precisely/ 42 integral points:
- --- In firstname.lastname@example.org, "djbroadhurst" <d.broadhurst@...> wrote:
>That was 9 months ago.
> > http://physics.open.ac.uk/~dbroadhu/cert/mwrank9.txt
> > is growing rather slowly
> This continues, with merely 18 curves currently in that rank-9 file.
> However, I hope that Mike may soon add to these, since in
> the case y^2 = x^3 + k with k < 0 his systematic methods
> may be more powerful than anything that Kevin or I have contrived.
> In any case, this continues to be a learning process for us,
> so thanks again to Cino:
> and then Robin:
> for getting us started.
Since then, this "learning process" by David, Kevin and me has continued, and if you visit that link you will find that those 18 elliptic curves have grown somewhat in number! And there are related pages, if you replace "9" by "8" thru "12" in the URL (that signifying the rank of the curves).
We have recently ventured into the rather scary territory of the Tate-Shafarevich group, which is currently not deeply understood by the number theory community. (It is a famous open problem to prove that it is always of finite order.)
You may be interested in our today's post to the NMBRTHRY list:
which describes a nontrivial result from our researches in this area.