Thread: any positive integer can be expressed..

1. any positive integer can be expressed..

Hello
I'm trying to prove the following conjecture,

Any positive integer $\displaystyle n$ can be expressed in the form
$\displaystyle n=\varepsilon _{1}1^2+\varepsilon _22^2+\varepsilon _33^2+...+\varepsilon _mm^2$
where $\displaystyle m$ a positive integer and $\displaystyle \varepsilon _i=1$ or $\displaystyle -1,i=1,2,3,...,m$.
Any ideas ?

2. It seems likely but difficult to prove. Did you make this up for fun?

3. $\displaystyle 3=-1^2+2^2$

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

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

$\displaystyle 4=-1^2-2^2+3^2$
Bah! I'm going to bed!

Thanks for the catch.

-Dan

5. @Bruno.J
No i didn't make anything up.

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

7. Originally Posted by Raoh
$\displaystyle 2 = -1^2-2^2 - 3^2 +4^2.$
$\displaystyle 5 = 1^2+2^2.$
$\displaystyle 6 = 1^2 - 2^2 + 3^2.$
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

8. 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