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Math Help - square of an integer

  1. #1
    Member disclaimer's Avatar
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    square of an integer

    Show that 2004^2+2004^2\cdot2005^2+2005^2 is the square of an integer.

    Any hint will be appreciated.
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by disclaimer View Post
    Show that 2004^2+2004^2\cdot2005^2+2005^2 is the square of an integer.

    Any hint will be appreciated.
    It can be proved in general :
    N^2+N^2 \cdot (N+1)^2+(N+1)^2 is a square.

    All you need to know is :
    (a-b)^2=a^2+b^2-2ab
    (a+b)^2=a^2+b^2+2ab


    ---------------------------------------------
    N^2+(N+1)^2=\underbrace{N^2+(N+1)^2-2N(N+1)}_{[(N+1)-N]^2}+2N(N+1)

    So N^2+N^2 \cdot (N+1)^2+(N+1)^2=[(N+1)-N]^2+2N(N+1)+[N(N+1)]^2

    =1+2N(N+1)+[N(N+1)]^2=\boxed{[N(N+1)+1]^2}

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  3. #3
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    Thank you, Moo.
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