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 August 29th, 2013, 05:53 PM #21 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 .... Both of those formulas, as written, are false. Perhaps you meant to sum up to something other than n-1?
August 29th, 2013, 10:56 PM   #22
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Re: Express a cube as sum of squares ....

Quote:
Originally Posted by icemanfan
Quote:
 Originally Posted by CRGreathouse You can express any nonnegative integer as a sum of four squares.
Do you know how to prove that? I would be interested in seeing the argument.
That is Lagrange's four square theorem. It is a well-known special case of the Waring problem.

August 30th, 2013, 12:41 AM   #23
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Re: Express a cube as sum of squares ....

Quote:
 Originally Posted by CRGreathouse (You could also use the WZ method, as found in the free (!) book A = B, but that would take much longer to learn!)
Is it this book ? http://www.math.upenn.edu/~wilf/AeqB.html

August 30th, 2013, 04:19 AM   #24
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Re: Express a cube as sum of squares ....

Quote:
 Originally Posted by CRGreathouse Both of those formulas, as written, are false. Perhaps you meant to sum up to something other than n-1?
Right.
I mistaped n instead of k assuming that n even is equal to 2k and n odd =2k+1
So you sum up to 2(k-1).

Edited twice

 August 30th, 2013, 04:22 AM #25 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... Correction 2 When n is even =2k n^3 = 6*(sigma(2i^2) with i=1 to 2(k-1)) +3n^2-2n When is n is odd =2k+1 n^3=6*(sigma((2i+1)^2) with i=0 to 2(k-1)) +3n^2-2n In Latex it will be great.
August 30th, 2013, 04:24 AM   #26
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Re: Express a cube as sum of squares ....

Quote:
Originally Posted by icemanfan
Quote:
 Originally Posted by Mouhaha Can you rewrite my formulas in Latex? It would help a lot I think.
For n even=2k

$n^3= 6\sum_{i=1}^{2(k-1)} 2i^2 + 3n^2 - 2n$

For n odd=2k+1

$n^3= 6\sum_{i=0}^{2(k-1)}(2i+1)^2 + 3n^2 - 2n$
Corrected twice

 August 30th, 2013, 04:26 AM #27 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... So nothing new. The formula seems to be known. Is it useful?
 August 30th, 2013, 04:36 AM #28 Member   Joined: Apr 2013 Posts: 70 Thanks: 0 Re: Express a cube as sum of squares .... Sorry!!!!!!!!!!!!!! Sum up to 2(k-1) not n-1
August 30th, 2013, 10:25 AM   #29
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Re: Express a cube as sum of squares ....

Quote:
Originally Posted by gelatine1
Quote:
 Originally Posted by CRGreathouse (You could also use the WZ method, as found in the free (!) book A = B, but that would take much longer to learn!)
Is it this book ? http://www.math.upenn.edu/~wilf/AeqB.html
Yes. I highly recommend it.

August 30th, 2013, 10:26 AM   #30
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Re: Express a cube as sum of squares ....

Quote:
 Originally Posted by Mouhaha So nothing new. The formula seems to be known. Is it useful?
I don't know of any applications for this particular formula, but there are certainly uses for the general method of turning the sum of a polynomial into a polynomial of higher degree.

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