November 3rd, 2008, 11:07 PM  #1 
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  A Diophantine equation
Please find a nonzero integer solution for equation 
November 5th, 2008, 07:54 PM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Diophantine equation
(1, 1, 2, 2, 2, 4  sqrt(2))

November 6th, 2008, 12:09 AM  #3  
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: A Diophantine equation Quote:
 
November 6th, 2008, 05:39 AM  #4 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Diophantine equation
Ah, missed that, sorry. I wasn't able to find any nonzero integer solutions.

November 6th, 2008, 05:43 AM  #5 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Diophantine equation
Mathematica 6.0 fails: Code: In[1]:= FindInstance[ a^2 + b^2 + c^2 + d^2 + e^2 + f^2 == a b c d e f && a > 0, {a, b, c, d, e, f}, Integers] During evaluation of In[1]:= FindInstance::nsmet: The methods \ available to FindInstance are insufficient to find the requested \ instances or prove they do not exist. >> Out[1]= FindInstance[ a^2 + b^2 + c^2 + d^2 + e^2 + f^2 == a b c d e f && a > 0, {a, b, c, d, e, f}, Integers] 
November 6th, 2008, 05:07 PM  #6 
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: A Diophantine equation
Hehe, How if there're 5 or 7 variables instead of 6 variables?

November 7th, 2008, 12:17 PM  #7 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 937 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: A Diophantine equation
It's can't find anything for 5. For 7, it gives (3, 2, 2, 2, 1, 1, 1).

November 7th, 2008, 03:25 PM  #8 
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: A Diophantine equation
So it seems it is not so good. For five numbers, below is a solution 1 1 3 3 4 
November 7th, 2008, 03:29 PM  #9 
Senior Member Joined: Oct 2008 Posts: 215 Thanks: 0  Re: A Diophantine equation
In fact, I have proved that there're no nonzero solutions for 6 variables with help of computer: See more details at http://zdu.spaces.live.com/blog/cns!...2037!164.entry 

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