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Math Help - any positive integer can be expressed..

  1. #1
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    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 ?
    Last edited by Raoh; February 27th 2011 at 02:55 PM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    It seems likely but difficult to prove. Did you make this up for fun?
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    3=-1^2+2^2

    4=-1^2-2^2+3^2
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Bruno J. View Post
    3=-1^2+2^2

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

    Thanks for the catch.

    -Dan
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  5. #5
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    @Bruno.J
    No i didn't make anything up.
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  6. #6
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    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 =)
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  7. #7
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    Quote Originally Posted by Raoh View Post
    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
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  8. #8
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    Quote Originally Posted by Wilmer View Post
    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
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  9. #9
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    Right...my bad...but:
    -1 - 4 + 9 = 4
    1 - 4 - 9 + 16 = 4

    My point? More than 1 solution...
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  10. #10
    MHF Contributor Bruno J.'s Avatar
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    This thread deserves to be moved in Number Theory!
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