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August 29th, 2013, 12:29 PM  #1 
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Express a cube as sum of squares ....
Hi there, Is there a way to express n^3 as sum of squares and n`s? n is an integer>1 Thank you. 
August 29th, 2013, 01:23 PM  #2 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Express a cube as sum of squares ....
You need to be more specific with this question. There is only one way of representing in terms of n, although it is possible that can be represented as a sum of squares. Trivially, every cube can be expressed as a sum of squares since 1 is a square. If you were to change this to "sum of distinct squares" then there would be many cubes which could not be so represented, 8 being the first. Another possibility is to restrict the number of squares. For instance, , so it is representable as a sum of two squares.

August 29th, 2013, 01:34 PM  #3 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Express a cube as sum of squares ....
You can express any nonnegative integer as a sum of four squares. Do you have something more specific in mind?

August 29th, 2013, 01:39 PM  #4  
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Express a cube as sum of squares .... Quote:
 
August 29th, 2013, 01:40 PM  #5 
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Express a cube as sum of squares ....
Sorry. I`m talking about a general formula. n^3 is expressed by n`s (squares and degree 1) 
August 29th, 2013, 01:42 PM  #6  
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Express a cube as sum of squares .... Quote:
 
August 29th, 2013, 01:44 PM  #7  
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Express a cube as sum of squares .... Quote:
 
August 29th, 2013, 03:23 PM  #8 
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Express a cube as sum of squares ....
please read it carefully how to express n^3 as elegant sum of squares and n`s? Let n=2k an even integer C(n,3)=(n*(n1)*(n2))/6 Here is an identity to check : C(n,3)=sigma(2i^2)) with i=1 to n1 Example : C(8,3)=2^2+4^2+6^2=56 Let n=2k+1 an odd integer Here is an identity to check : C(n,3)=sigma((2i+1)*^2) with i=0 to n1 Example : C(11,3)=1^2+3^2+5^2+7^2+9^2=165 Now you can easily deduce n^3 and express it as said above. I will let you do it and write it in Latex. 
August 29th, 2013, 03:52 PM  #9 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Express a cube as sum of squares ....
If you're allowed unbounded sums there are thousands of identities. You can check this one with Faulhaber's formula, if you like.

August 29th, 2013, 03:54 PM  #10  
Member Joined: Apr 2013 Posts: 70 Thanks: 0  Re: Express a cube as sum of squares .... Quote:
 

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