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Math Help - Always square

  1. #1
    Junior Member theprof's Avatar
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    Always square


    1x2x3x4 + 1 = 25 = 5^2
    3x4x5x6 + 1 = 361 = 19^2
    ...


    Prove that the product of 4 whole consecutive numbers plus one is always a perfect square.

    bye
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  2. #2
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    If by whole numbers you mean Z+,

    For any whole number n,

    n*(n+1)*(n+2)*(n+3)+1
    = n^4+6*n^3+11*n^2+6*n+1
    = (n^2+3*n+1)^2

    (n^2+3*n+1)^2 is the square of (n^2+3*n+1), and (n^2+3*n+1) is an integer.
    Last edited by paultwang; April 11th 2005 at 02:00 PM.
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  3. #3
    Junior Member theprof's Avatar
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    Quote Originally Posted by paultwang
    If by whole numbers you mean Z+,
    As far as I can see (n^2+3n+1)^2 is a perfect square for all n \in Z, not only Z+...
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