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Math Help - Show that it is an integer

  1. #1
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    Show that it is an integer

    Show that for any a, b, c different integers:
    \frac {a^{2008}}{(a-b)(a-c)}+\frac {b^{2008}}{(b-a)(b-c)}+\frac {c^{2008}}{(c-b)(c-a)}\in\mathbb{Z}
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  2. #2
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    Quote Originally Posted by james_bond View Post
    Show that for any a, b, c different integers:
    \frac {a^{2008}}{(a-b)(a-c)}+\frac {b^{2008}}{(b-a)(b-c)}+\frac {c^{2008}}{(c-b)(c-a)}\in\mathbb{Z}
    I don't see a neat way to do this. The only method that I could find is to put it over a common denominator, to get

    -\dfrac{a^n(b-c)+b^n(c-a) + c^n(a-b)}{(b-c)(c-a)(a-b)} = -\dfrac1{(b-c)(c-a)(a-b)} \begin{vmatrix}a^n&b^n&c^n\\ a&b&c\\ 1&1&1\end{vmatrix}.

    Then doing row and column operations on the determinant I found that it is equal to -(b-c)(c-a)(a-b)\sum a^ib^jc^k, where the sum is taken over all triples (i,j,k) of nonnegative integers such that i+j+k=n-2.
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