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$Old$ 10-25-2008, 12:51 AM

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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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$Old$ 10-25-2008, 01:45 AM

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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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