Cubes which are Sum of Three Cubes

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Theorem

The following cube numbers can be expressed as the sum of $3$ positive cube numbers:

$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$

This sequence is A066890 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The associated cube roots:

$6, 9, 12, 18, 19, 20, 24, 25, \ldots$

This sequence is A023042 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Examples

\(\ds 6^3\) \(=\) \(\ds 216\)
\(\ds \) \(=\) \(\ds 27 + 64 + 125\)
\(\ds \) \(=\) \(\ds 3^3 + 4^3 + 5^3\)


\(\ds 9^3\) \(=\) \(\ds 729\)
\(\ds \) \(=\) \(\ds 1 + 216 + 512\)
\(\ds \) \(=\) \(\ds 1^3 + 6^3 + 8^3\)


Sources

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