Cube which can be Represented as Sum of 3, 4, 5, 6, 7 or 8 Cubes

Theorem

 * $351 \, 120^3$ can be represented as the sum of $3$, $4$, $5$, $6$, $7$ or $8$ cubes.

Proof
These representations are not necessarily unique.

Additional Results
We also have:

So $351120^3$ can also be expressed as a sum of $9$ or $10$ cubes.

These equations all stem from:

showing that $351 \, 120$ is not the smallest number with this property.

Moreover, using $3^3 + 4^3 + 5^3 = 6^3$, this result could still be further extended:

which is a sum of $11$ to $19$ cubes.

Using $1^3 + 6^3 + 8^3 = 9^3$ and $\paren {2 n}^3 = 8 \times n^3$, we can express $12^3$ as more and more cubes.