Euler (6,2,5) details.

Death · 2010-04-30, 10:34

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.

R. Gerbicz · 2010-06-11, 18:03

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.

kar_bon · 2010-06-11, 18:16

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

R.D. Silverman · 2010-06-11, 18:29

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.......

frmky · 2010-06-11, 23:31

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.

yoyo · 2010-06-22, 18:30

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

henryzz · 2010-06-22, 18:56

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

what about win64?

yoyo · 2010-06-22, 19:02

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

R. Gerbicz · 2011-08-03, 01:04

Our second project has been finished. See an article about it: http://arxiv.org/abs/1108.0462

CRGreathouse · 2011-08-03, 03:13

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?

R.D. Silverman · 2011-08-03, 13:49

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.

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.

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2010-06-11, 18:16	#3
kar_bon Mar 2006 Germany 2³·3·11² Posts	Congrats! Some years ago I've contibuted there, too.

2010-06-22, 18:30	#6
yoyo Oct 2006 Berlin, Germany 617 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, 19:02	#8
yoyo Oct 2006 Berlin, Germany 617₁₀ 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 10111001100₂ 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 3·1,993 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?