Integer as Sum of 4 Cubes

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

$23 = 8 + 8 + 8 - 1 = 2^3 + 2^3 + 2^3 + \paren {-1}^3$

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): $23$