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 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.