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:

\(\ds 23\) \(=\) \(\ds 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3\)
\(\ds 239\) \(=\) \(\ds 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3\)


Proof


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In particular: It needs to be demonstrated that these are the only ones.
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Sources

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