Euler 625 Details
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- Mikrocruncher
- Beiträge: 27
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Euler 625 Details
I'm the author of the original project about computing (6,2,5): http://euler.free.fr/
The project is running since 11 years, but I think the new yoyo project will outperform our project in a few days.
The goal is to compute solutions to the equation:
a^6 + b^6 = c^6 + d^6 + e^6 + f^6 + g^6
We use the notation (6,2,5) to express the fact that the equation is at 6th power, has 2 left terms and 5 right terms.
Reminder: ^ means power. Thus a^6 = a*a*a*a*a*a.
For example, 40^6 = 4096000000, as you can see, a^6 grows very quickly.
Why is this equation particularly interesting ?
In fact, it is not very useful.
We hope to find a solution where one of the terms a,b,c,d,e,f or g is zero, in other words, this means that we are searching (6,1,5) or (6,2,4).
Why are we searching for (6,1,5) or (6,2,4) ?
Some mathematicians conjectured that we could find solutions of equations (k,m,n), where k=m+n.
In other words, we can find a combination of 6 terms at the 6th power that lead to 0 when added or subtracted.
Currently, we found solutions for the following equations:
(4,1,3), (4,2,2)
(5,1,4) (5,2,3)
(6,3,3)
(8,3,5) (8,4,4)
For example, for (6,3,3), Subba-Rao found the following result in 1934:
23^6+15^6+10^6=22^6+19^6+3^6
in fact, there is an infinite number of solutions for (6,3,3)
To extend the above list of results, the easier solutions to reach are, in increasing difficulty order:
(6,2,4), (7,3,4), (7,2,5), (7,1,6), (8,2,6), (8,1,7)
Searching for (6,1,5) is not very useful, because we don't even know a solution to (6,1,6) !
Since I started this project 11 years ago, a lot of results were discovered, and quite a large amount of CPU has been spent onto them.
New results keep coming, but there is no real breakthrough.
Now, about the current Euler 625 project.
The program will compute solutions such that max(a,b,c,d,e,f,g)=7^6=117649
This may seem not very impressive, but don't forget that we will find solutions where the left or right numbers will be below
2*(117649^6)=5303461691719306943558046763202=5.303461691719306943558046763202e+30 !
or numbers using 103 bits.
My old project is able to compute (6,2,5) but it's very slow, compared to the algorithm used into this project.
The new project will help us double-check our old solutions (to check if we didn't miss one), and will probably find new ones.
I'll keep you informed here when a new solution is discovered.
Note also that the program discovers a lot of solutions where all the numbers are divisible by a small value (like 2, 3 or 5). We call these solutions not primitive, since a solution is still valid after a multiplication by any integer value !
Finally, I'd like to thank several people:
- Yoyo, because he setup the whole project. I hope we'll find new solutions very soon.
- Robert Gerbicz, who designed the new algorithm. Without him, we could not run this search
- Dead J. Dona for suggesting to start this project
- Greg Childers, who maintained the server for the old project during all these years, and finding new approaches.
Ok, I think that's all.
Good luck on this search !
The project is running since 11 years, but I think the new yoyo project will outperform our project in a few days.
The goal is to compute solutions to the equation:
a^6 + b^6 = c^6 + d^6 + e^6 + f^6 + g^6
We use the notation (6,2,5) to express the fact that the equation is at 6th power, has 2 left terms and 5 right terms.
Reminder: ^ means power. Thus a^6 = a*a*a*a*a*a.
For example, 40^6 = 4096000000, as you can see, a^6 grows very quickly.
Why is this equation particularly interesting ?
In fact, it is not very useful.
We hope to find a solution where one of the terms a,b,c,d,e,f or g is zero, in other words, this means that we are searching (6,1,5) or (6,2,4).
Why are we searching for (6,1,5) or (6,2,4) ?
Some mathematicians conjectured that we could find solutions of equations (k,m,n), where k=m+n.
In other words, we can find a combination of 6 terms at the 6th power that lead to 0 when added or subtracted.
Currently, we found solutions for the following equations:
(4,1,3), (4,2,2)
(5,1,4) (5,2,3)
(6,3,3)
(8,3,5) (8,4,4)
For example, for (6,3,3), Subba-Rao found the following result in 1934:
23^6+15^6+10^6=22^6+19^6+3^6
in fact, there is an infinite number of solutions for (6,3,3)
To extend the above list of results, the easier solutions to reach are, in increasing difficulty order:
(6,2,4), (7,3,4), (7,2,5), (7,1,6), (8,2,6), (8,1,7)
Searching for (6,1,5) is not very useful, because we don't even know a solution to (6,1,6) !
Since I started this project 11 years ago, a lot of results were discovered, and quite a large amount of CPU has been spent onto them.
New results keep coming, but there is no real breakthrough.
Now, about the current Euler 625 project.
The program will compute solutions such that max(a,b,c,d,e,f,g)=7^6=117649
This may seem not very impressive, but don't forget that we will find solutions where the left or right numbers will be below
2*(117649^6)=5303461691719306943558046763202=5.303461691719306943558046763202e+30 !
or numbers using 103 bits.
My old project is able to compute (6,2,5) but it's very slow, compared to the algorithm used into this project.
The new project will help us double-check our old solutions (to check if we didn't miss one), and will probably find new ones.
I'll keep you informed here when a new solution is discovered.
Note also that the program discovers a lot of solutions where all the numbers are divisible by a small value (like 2, 3 or 5). We call these solutions not primitive, since a solution is still valid after a multiplication by any integer value !
Finally, I'd like to thank several people:
- Yoyo, because he setup the whole project. I hope we'll find new solutions very soon.
- Robert Gerbicz, who designed the new algorithm. Without him, we could not run this search
- Dead J. Dona for suggesting to start this project
- Greg Childers, who maintained the server for the old project during all these years, and finding new approaches.
Ok, I think that's all.
Good luck on this search !
Zuletzt geändert von jcmeyrignac am 16.05.2010 12:59, insgesamt 1-mal geändert.
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- Projekt-Fetischist
- Beiträge: 639
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Re: Euler 625 Details
The question i have is, why don't we search for 6,2,4 directly? Wouldn't it be faster?
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- Taschenrechner
- Beiträge: 11
- Registriert: 14.04.2010 08:32
Re: Euler 625 Details
Jean,
that was Yoyo who made a binary for windows.
I'm just an enthusiast who find the right people to make this project possible and gather them.
All hails to you, Yoyo and Robert.
that was Yoyo who made a binary for windows.
I'm just an enthusiast who find the right people to make this project possible and gather them.
All hails to you, Yoyo and Robert.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
From my previous benchmarks, searching for 6,2,4 is marginally faster, but there is a good chance that we won't find it.Thommy3 hat geschrieben:The question i have is, why don't we search for 6,2,4 directly? Wouldn't it be faster?
I believe it's more gratifying to find solutions rather than nothing.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
So far, two new solutions have been discovered (I use my own notation here):
(6,2,5) 92711+47567=83027+80556+59802+14700+14029
(6,2,5) 95713+63016=91080+79423+46074+9646+3402
We can expect 30 more solutions before the end of this computation.
And we can hope for a (6,2,4)...
(6,2,5) 92711+47567=83027+80556+59802+14700+14029
(6,2,5) 95713+63016=91080+79423+46074+9646+3402
We can expect 30 more solutions before the end of this computation.
And we can hope for a (6,2,4)...
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- Vereinsvorstand
- Beiträge: 8048
- Registriert: 17.12.2002 14:09
- Wohnort: Berlin
Re: Euler 625 Details
Thanks for the update. I posted it into the news section on the home page.
yoyo
yoyo
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
Another new solution:
107726^6+84205^6=110358^6+59872^6+59437^6+48342^6+46410^6
or using my notation:
(6,2,5) 107726+84205=110358+59872+59437+48342+46410
So far, 51 primitive solutions have been discovered (primitive means that we divided the solutions by their GCD).
152 solutions are currently known.
107726^6+84205^6=110358^6+59872^6+59437^6+48342^6+46410^6
or using my notation:
(6,2,5) 107726+84205=110358+59872+59437+48342+46410
So far, 51 primitive solutions have been discovered (primitive means that we divided the solutions by their GCD).
152 solutions are currently known.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
New solution:
100789^6+5599^6=98115^6+26244^6+30786^6+42896^6+72793^6
(6,2,5) 100789+5599=98115+72793+42896+30786+26244
57 (out of 153) primitive solutions until now.
100789^6+5599^6=98115^6+26244^6+30786^6+42896^6+72793^6
(6,2,5) 100789+5599=98115+72793+42896+30786+26244
57 (out of 153) primitive solutions until now.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
No new solution, but we reached 70 solutions to (6,2,5), out of the 153 known.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
Another new solution discovered by Liuqyn and proteino.de:
97159^6+64893^6=89222^6+83967^6+57141^6+54474^6+5490^6
or:
(6,2,5) 97159+64893=89222+83967+57141+54474+5490
We reached 76 solutions, out of 154 known.
97159^6+64893^6=89222^6+83967^6+57141^6+54474^6+5490^6
or:
(6,2,5) 97159+64893=89222+83967+57141+54474+5490
We reached 76 solutions, out of 154 known.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
New solution, discovered by Jim PROFIT and Greeri
107035^6+52081^6=105847^6+65112^6+58758^6+9555^6+2482^6
(6,2,5) 107035+52081=105847+65112+58758+9555+2482
Current status of the computation: 85 solutions out of 155 known.
107035^6+52081^6=105847^6+65112^6+58758^6+9555^6+2482^6
(6,2,5) 107035+52081=105847+65112+58758+9555+2482
Current status of the computation: 85 solutions out of 155 known.
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- Mikrocruncher
- Beiträge: 27
- Registriert: 29.04.2010 12:18
Re: Euler 625 Details
3 new solutions today:
(6,2,5) 100755+63397=93338+82068+66612+62019+20469 by MAVRIK and Greeri
(6,2,5) 101159+42058=97678+74046+59367+6760+6534 by rroonnaalldd and Pwrguru
(6,2,5) 114361+12374=102552+98658+65977+64036+24528 by proteino.de and [AF>HFR>RR] julien76100
We reached 101 out of 158 known solutions.
(6,2,5) 100755+63397=93338+82068+66612+62019+20469 by MAVRIK and Greeri
(6,2,5) 101159+42058=97678+74046+59367+6760+6534 by rroonnaalldd and Pwrguru
(6,2,5) 114361+12374=102552+98658+65977+64036+24528 by proteino.de and [AF>HFR>RR] julien76100
We reached 101 out of 158 known solutions.