# Numbers not Expressible as Sum of Less than 9 Positive Cubes

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## Theorem

The following are the only positive integers cannot be expressed as the sum of less than $9$ positive cubes:

\(\displaystyle 23\) | \(=\) | \(\displaystyle 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3\) | |||||||||||

\(\displaystyle 239\) | \(=\) | \(\displaystyle 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3\) |

## Proof

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $23$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $239$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $23$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $239$