Prove:

**great_math** · October 24th 2008, 10:51 PM

Prove that $a^3+b^3+c^3+d^3+e^3= n$ has an integer solution for any intger n.

**NonCommAlg** · October 24th 2008, 11:45 PM

Originally Posted by great_math

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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