- --- In primenumbers@yahoogroups.com, Robin Garcia <sopadeajo2001@...> wrote:
>

[snip]

> Some small results for cubes and squares:

>

> It seems, checked to t=10^7, but probably true for any powers, there is never a difference of 3, of 5 , of 6, of 10, of 14, of 16 (below 20) between cubes and squares but a better experimental confirmation is expected. And a proof would be nice.

Hi Robin,

What about

1^3=1 2^2=4 delta=-3 ?

> d = 17 : 6 ways

[Not sure what your "checked to t=10^7" means.]

>

> 5^2 = 2^3+17

>

> 9^2 = 4^3+17

>

> 23^2 = 8^3+17

>

> 282^2 = 43^3+17

>

> 375^2 = 52^3+17

>

> 378661^2 = 5234^3+17

>

> Wellcome to all those who will try to find a 7 ways result !!

Defining delta=n^3-m^2, I have investigated abs(delta)<=400 for n <= 10^8.

There are just two more-than-6-way delta's:-

delta=207: 7 ways

6^3=216 3^2=9 delta=207

12^3=1728 39^2=1521 delta=207

18^3=5832 75^2=5625 delta=207

31^3=29791 172^2=29584 delta=207

312^3=30371328 5511^2=30371121 delta=207

331^3=36264691 6022^2=36264484 delta=207

367806^3=49757256774842616 223063347^2=49757256774842409 delta=207

delta=-225: 10 ways

4^3=64 17^2=289 delta=-225

6^3=216 21^2=441 delta=-225

10^3=1000 35^2=1225 delta=-225

15^3=3375 60^2=3600 delta=-225

30^3=27000 165^2=27225 delta=-225

60^3=216000 465^2=216225 delta=-225

180^3=5832000 2415^2=5832225 delta=-225

336^3=37933056 6159^2=37933281 delta=-225

351^3=43243551 6576^2=43243776 delta=-225

720114^3=373425320872841544 611085363^2=373425320872841769 delta=-225

Mike - --- In primenumbers@yahoogroups.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:

> http://tech.groups.yahoo.com/group/primenumbers/message/20404

> and then Robin:

> http://tech.groups.yahoo.com/group/primenumbers/message/23589

> for getting us started.

>

> David

>

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:

https://listserv.nodak.edu/cgi-bin/wa.exe?A2=nmbrthry;8f3b553f.1208

which describes a nontrivial result from our researches in this area.

Mike