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 2010-04-30, 10:34 #1 Death     Mar 2004 Ukraine, Kiev 2×23 Posts Euler (6,2,5) details. Good day. You all know this great site http://euler.free.fr/ 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: a6 + b6 = c6 + d6 + e6 + f6 + g6 More detailed explanation by Jean-Charles Meyrignac can be found here. I won't repost entire topic, if anybody desired to do this, feel free to do so.
2010-06-11, 18:03   #2
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

2×733 Posts

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.
Attached Files
 known.txt (9.8 KB, 173 views)

 2010-06-11, 18:16 #3 kar_bon     Mar 2006 Germany 23·192 Posts Congrats! Some years ago I've contibuted there, too.
2010-06-11, 18:29   #4
R.D. Silverman

Nov 2003

1D2416 Posts

Quote:
 Originally Posted by R. Gerbicz 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.
Now all we have to do is find a (6,1,5) and (7,1,6) solution.......

2010-06-11, 23:31   #5
frmky

Jul 2003
So Cal

2,089 Posts

Quote:
 Originally Posted by R.D. Silverman Now all we have to do is find a (6,1,5) and (7,1,6) solution.......
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.

 2010-06-22, 18:30 #6 yoyo     Oct 2006 Berlin, Germany 22×32×17 Posts 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
2010-06-22, 18:56   #7
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

10110111011012 Posts

Quote:
 Originally Posted by yoyo 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

 2010-06-22, 19:02 #8 yoyo     Oct 2006 Berlin, Germany 22·32·17 Posts 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
 2011-08-03, 01:04 #9 R. Gerbicz     "Robert Gerbicz" Oct 2005 Hungary 2×733 Posts Our second project has been finished. See an article about it: http://arxiv.org/abs/1108.0462
 2011-08-03, 03:13 #10 CRGreathouse     Aug 2006 32×5×7×19 Posts Coincidentally, I was just looking at Euler(2..5, 1, k) equations yesterday -- in particular Sloane's A161882, A161883, A161884, and A161885 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?
2011-08-03, 13:49   #11
R.D. Silverman

Nov 2003

22×5×373 Posts

Quote:
 Originally Posted by CRGreathouse Coincidentally, I was just looking at Euler(2..5, 1, k) equations yesterday -- in particular Sloane's A161882, A161883, A161884, and A161885 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?
The one to ask would be Bob Vaughn. I believe that he is at Penn State.

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