- 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.

Differences are expressable in several ways :

d=1 : 1 way, proved there is only one way and for any power, not just squares and cubes; by a romanian.

3^2 = 2^3+1

d = 8 : 2 ways

6^2 = 3^3+8

312^2 = 46^3+8

d=11 :

4^2 = 3^3-11

58^2 = 15^3-11

d = 174

13^2 = 7^3-174

22585^2 = 799^3-174

(These last two for d=11 and 174 just to show negative differences also give results)

d = 9 : 3 ways

6^2 = 3^3+9

15^2 = 6^3+9

253^2 = 40^3+9

d = 73 : 4 ways

10^2 = 3^3+73

17^2 = 6^3+73

611^2 = 72^3+73

6717^2 = 356^3+73

d = 17 : 6 ways

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 !!

Note: This has been done in a hurry. Results could certainly be incomplete.

[Non-text portions of this message have been removed] - --- 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