20100430, 10:34  #1 
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: a^{6} + b^{6} = c^{6} + d^{6} + e^{6} + f^{6} + g^{6} More detailed explanation by JeanCharles Meyrignac can be found here. I won't repost entire topic, if anybody desired to do this, feel free to do so. 
20100611, 18:03  #2 
"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. 
20100611, 18:16  #3 
Mar 2006
Germany
2^{3}·19^{2} Posts 
Congrats! Some years ago I've contibuted there, too.

20100611, 18:29  #4 
Nov 2003
1D24_{16} Posts 
Now all we have to do is find a (6,1,5) and (7,1,6) solution.......

20100611, 23:31  #5 
Jul 2003
So Cal
2,089 Posts 

20100622, 18:30  #6 
Oct 2006
Berlin, Germany
2^{2}×3^{2}×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 
20100622, 18:56  #7 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
1011011101101_{2} Posts 

20100622, 19:02  #8 
Oct 2006
Berlin, Germany
2^{2}·3^{2}·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 
20110803, 01:04  #9 
"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

20110803, 03:13  #10 
Aug 2006
3^{2}×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 foursquare 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? 
20110803, 13:49  #11  
Nov 2003
2^{2}×5×373 Posts 
Quote:


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