# Integer as Sum of 4 Cubes

Jump to navigation
Jump to search

## 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

## 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): $23$