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August 31st, 2013, 12:51 PM   #31
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Re: Express a cube as sum of squares ....

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Originally Posted by CRGreathouse
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.
What about using it to disprove the FLT a^3+b^3=c^3 ?
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August 31st, 2013, 06:10 PM   #32
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Re: Express a cube as sum of squares ....

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Originally Posted by Mouhaha
What about using it to disprove the FLT a^3+b^3=c^3 ?
I don't see a way to do that with this (simple) formula.
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September 3rd, 2014, 11:34 AM   #33
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"You can express any nonnegative integer as a sum of four squares. "

This seems incorrect. Certainly said integer would have to be at least 4, but I can't fathom any four squares that add to 5, 6, 8, 9... I think I could go on.
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September 3rd, 2014, 11:45 AM   #34
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Quote:
Originally Posted by DaneBrooke View Post
"You can express any nonnegative integer as a sum of four squares. "

This seems incorrect. Certainly said integer would have to be at least 4, but I can't fathom any four squares that add to 5, 6, 8, 9... I think I could go on.
$5=0^2+0^2+1^2+2^2$

If you want numbers that are the sum of four nonzero squares you're looking for
https://oeis.org/A000414
rather than the nonnegative integers.
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September 3rd, 2014, 03:23 PM   #35
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Hello, Mouhaha!

Quote:
Is there a way to express n^3 as sum of squares and n's?
n is an integer > 1.
I don't know what you mean by "and n's."

I assume you mean a sum of distinct squares.

Otherwise, we have these trivial solutions:

$\qquad \begin{array}{ccc}2^3 &=& 2^2+2^2 \\ 3^3 &=& 3^2+3^2+3^2 \\ 4^3 &=& 4^2+4^2+4^2+4^2 \\ \vdots && \vdots \end{array}$


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