Re: Primes again [really Mordell's equation]
--- In email@example.com, "djbroadhurst" <d.broadhurst@...> wrote:
> shows the origin of my Prime Puzzle 36.
> > Second hint: Jens Kruse Andersen has already high-lighted
> > Donovan Johnson's prime, in a closely related Prime Puzzle.
> show a cognate Prime Puzzle, illuminated by Jens.
Results from a neat little algorithm devised by David Broadhurst has allowed extension of the above puzzle.
In particular the prime number 107122676734733201 is a combination of positive cubes and squares in 14 different ways.
David's investigations also unearthed a couple of primes (4417190430889897 and 84658174289284249), originally found in a search by Noam Elkies, that are combinations of positive cubes and squares in 11 and 12 different ways respectively.
is now updated with these three new entries.
Many thanks to David for both his algorithmic ingenuity and scholarly archeology.
- --- 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.