# any positive integer can be expressed..

• Feb 27th 2011, 01:32 PM
Raoh
any positive integer can be expressed..
Hello
I'm trying to prove the following conjecture,

Any positive integer $n$ can be expressed in the form
$n=\varepsilon _{1}1^2+\varepsilon _22^2+\varepsilon _33^2+...+\varepsilon _mm^2$
where $m$ a positive integer and $\varepsilon _i=1$ or $-1,i=1,2,3,...,m$.
Any ideas ?
• Feb 27th 2011, 07:51 PM
Bruno J.
It seems likely but difficult to prove. Did you make this up for fun?
• Feb 27th 2011, 08:01 PM
Bruno J.
$3=-1^2+2^2$

$4=-1^2-2^2+3^2$
• Feb 27th 2011, 08:07 PM
topsquark
Quote:

Originally Posted by Bruno J.
$3=-1^2+2^2$

$4=-1^2-2^2+3^2$

Bah! I'm going to bed!

Thanks for the catch. :)

-Dan
• Feb 27th 2011, 09:49 PM
Raoh
@Bruno.J
No i didn't make anything up.
• Feb 28th 2011, 12:06 AM
Raoh
$2 = -1^2-2^2 - 3^2 +4^2.$
$5 = 1^2+2^2.$
$6 = 1^2 - 2^2 + 3^2.$
i've found a hint, i will come to discuss it later.I'm off =)
• Feb 28th 2011, 12:33 AM
Wilmer
Quote:

Originally Posted by Raoh
$2 = -1^2-2^2 - 3^2 +4^2.$
$5 = 1^2+2^2.$
$6 = 1^2 - 2^2 + 3^2.$
i've found a hint, i will come to discuss it later.I'm off =)

2 = -1^2 - 1^2 + 2^2
5 = -2^2 + 3^2
6 = 1^2 + 1^2 + 2^2
• Feb 28th 2011, 09:22 AM
Raoh
Quote:

Originally Posted by Wilmer
2 = -1^2 - 1^2 + 2^2
5 = -2^2 + 3^2
6 = 1^2 + 1^2 + 2^2

Yes,but that's not what the conjecture states :)
• Feb 28th 2011, 09:37 AM
Wilmer
-1 - 4 + 9 = 4
1 - 4 - 9 + 16 = 4

My point? More than 1 solution...
• Feb 28th 2011, 11:31 AM
Bruno J.
This thread deserves to be moved in Number Theory!