Re: Primes again [really Mordell's equation]
- --- In firstname.lastname@example.org,
"John" <mistermac39@...> wrote:
> > No integral point Q on E(-27*k) resultsThere are fancy names like injection and surjection
> > from a rational point P on E(k)
> I suspect this means there is no mapping, and one to one
> or any other correspondences
that I tend to get the wrong way round. I think the
right way round is to say that the map is injective,
but not surjective.
My heathen way of putting it is to say that every Q is mapped to
by at most one P. But there are lots of Q's that are mapped to
by no P's. And in the present case those Q's not given by P's
include all the integral points that form Mike's data.
At first I found this rather annoying. But then I realized that
it provides a wonderful blind check, whose validity is widely
believed, but not yet proven at the million-dollar mark.
- --- In email@example.com, "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.