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    Member great_math's Avatar
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    Prove:

    Prove that a^3+b^3+c^3+d^3+e^3= n has an integer solution for any intger n.
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  2. #2
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    Quote Originally Posted by great_math View Post

    Prove that a^3+b^3+c^3+d^3+e^3= n has an integer solution for any intger n.
    this is a very well-known problem! first note that 6 \mid n^3 - n, \ \forall n \in \mathbb{Z}. now we have: n=n^3 + \left(-1-\frac{n^3-n}{6} \right)^3 + \left(1 - \frac{n^3 - n}{6} \right)^3 + \left(\frac{n^3 -n }{6} \right)^3 + \left(\frac{n^3 -n }{6} \right)^3. \ \ \ \Box
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