Euler (6,2,5) details.
 

Good day.

You all know this great site [url]http://euler.free.fr/[/url] dedicated to Computing Minimal Equal Sums Of Like Powers.

Now BOINC project yoyo@home start to search for solutions of euler(6,2,5).

The goal is to compute solutions to the equation: a[SUP]6[/SUP] + b[SUP]6[/SUP] = c[SUP]6[/SUP] + d[SUP]6[/SUP] + e[SUP]6[/SUP] + f[SUP]6[/SUP] + g[SUP]6[/SUP]

More detailed explanation by Jean-Charles Meyrignac can be found [URL="http://www.rechenkraft.net/phpBB/viewtopic.php?f=57&t=10875"]here[/URL].

I won't repost entire topic, if anybody desired to do this, feel free to do so.

The project has been finished. Found 32 new solutions of the euler(6,2,5) system, and confirmed the previously known 149 solutions. In the attached known.txt file you can find the all known 181 (primitive) solutions.

I've written the c code that Boinc used.

Congrats! Some years ago I've contibuted there, too.

[QUOTE=R. Gerbicz;218223]The project has been finished. Found 32 new solutions of the euler(6,2,5) system, and confirmed the previously known 149 solutions. In the attached known.txt file you can find the all known 181 (primitive) solutions.

I've written the c code that Boinc used.[/QUOTE]

Now all we have to do is find a (6,1,5) and (7,1,6) solution.......

[QUOTE=R.D. Silverman;218231]Now all we have to do is find a (6,1,5) and (7,1,6) solution.......[/QUOTE]

The real goal of this search was to find a (6,2,4), which didn't happen. Even a (6,1,6) would be nice. :smile:

The Euler(6,2,5) project continues now with an increased range. We have applications for win32, linux32, linux64, Intel 32 Mac, Intel 64 Mac and PPC Mac available and are working on a Spar Solaris version.
yoyo

[quote=yoyo;219539]The Euler(6,2,5) project continues now with an increased range. We have applications for win32, linux32, linux64, Intel 32 Mac, Intel 64 Mac and PPC Mac available and are working on a Spar Solaris version.
yoyo[/quote]
what about win64?

I don't have a win64 system to compile the Boinc libs and the app there. I asked a team member to do it, but it will need some time.
yoyo

Our second project has been finished. See an article about it: [URL="http://arxiv.org/abs/1108.0462"]http://arxiv.org/abs/1108.0462[/URL]

Coincidentally, I was just looking at Euler(2..5, 1, k) equations yesterday -- in particular Sloane's [url=https://oeis.org/A161882]A161882[/url], [url=https://oeis.org/A161883]A161883[/url], [url=https://oeis.org/A161884]A161884[/url], and [url=https://oeis.org/A161885]A161885[/url] which look for the minimal k for a given n.

It can be shown from a reduction from known Waring numbers that any n has a nontrivial Euler(2, 1, k) solution with k <= 5, a nontrivial Euler(3, 1, k) solution with k <= 8, a nontrivial Euler(4, 1, k) solution with k <= 17, and a nontrivial Euler(5, 1, k) solution with k <= 38. In the first case the result can be improved with Jacobi's four-square theorem: there are multiple Euler(2, 1, 4) solutions for any n, so in particular at least one nontrivial solution. Does anyone know if that can be generalized? Usually I'd expect a lot of solutions so it doesn't seem like too much to ask.

Oh, and on the Euler(5, 1, k) problem I used g(5) rather than G(5) lacking information on the number of solutions needing more than, say, 17 summands -- the current bound on G(5). Does anyone have information on this?

[QUOTE=CRGreathouse;268156]Coincidentally, I was just looking at Euler(2..5, 1, k) equations yesterday -- in particular Sloane's [url=https://oeis.org/A161882]A161882[/url], [url=https://oeis.org/A161883]A161883[/url], [url=https://oeis.org/A161884]A161884[/url], and [url=https://oeis.org/A161885]A161885[/url] which look for the minimal k for a given n.

It can be shown from a reduction from known Waring numbers that any n has a nontrivial Euler(2, 1, k) solution with k <= 5, a nontrivial Euler(3, 1, k) solution with k <= 8, a nontrivial Euler(4, 1, k) solution with k <= 17, and a nontrivial Euler(5, 1, k) solution with k <= 38. In the first case the result can be improved with Jacobi's four-square theorem: there are multiple Euler(2, 1, 4) solutions for any n, so in particular at least one nontrivial solution. Does anyone know if that can be generalized? Usually I'd expect a lot of solutions so it doesn't seem like too much to ask.

Oh, and on the Euler(5, 1, k) problem I used g(5) rather than G(5) lacking information on the number of solutions needing more than, say, 17 summands -- the current bound on G(5). Does anyone have information on this?[/QUOTE]

The one to ask would be Bob Vaughn. I believe that he is at Penn State.