Integer as Sum of 4 Cubes
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Theorem
Let $n \in \Z$ be an integer.
Let $n \not \equiv 4 \pmod 9$ and $n \not \equiv 5 \pmod 9$.
Then it is possible to express $n$ as the sum of no more than $4$ cubes which may be either positive or negative.
Proof
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Examples
$23$ as Sum of $4$ Cubes
- $23 = 8 + 8 + 8 - 1 = 2^3 + 2^3 + 2^3 + \left({-1}\right)^3$
Also see
- Compare with the Hilbert-Waring theorem for $k = 3$: if the cubes all have to be positive then as many as $9$ may be needed.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $23$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $23$
