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

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Theorem

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


Proof

\(\ds 351120^3\) \(=\) \(\ds 175560^3 + 234080^3 + 292600^3\)
\(\ds \) \(=\) \(\ds 2 \times 87780^3 + 204820^3 + 321860^3\)
\(\ds \) \(=\) \(\ds 2 \times 87780^3 + 175560^3 + 2 \times 263340^3\)
\(\ds \) \(=\) \(\ds 3 \times 117040^3 + 3 \times 234080^3\)
\(\ds \) \(=\) \(\ds 2 \times 58520^3 + 117040^3 + 3 \times 175560^3 + 292600^3\)
\(\ds \) \(=\) \(\ds 8 \times 175560^3\)

These representations are not necessarily unique.

$\blacksquare$


Additional Results

We also have:

\(\ds 351120^3\) \(=\) \(\ds 58520^3 + 2 \times 117040^3 + 5 \times 175560^3 + 234080^3\)
\(\ds \) \(=\) \(\ds 5 \times 87780^3 + 4 \times 175560^3 + 263340^3\)

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


These equations all stem from:

\(\ds 12^3\) \(=\) \(\ds 6^3 + 8^3 + 10^3\)
\(\ds \) \(=\) \(\ds 2 \times 3^3 + 7^3 + 11^3\)
\(\ds \) \(=\) \(\ds 2 \times 3^3 + 6^3 + 2 \times 9^3\)
\(\ds \) \(=\) \(\ds 3 \times 4^3 + 3 \times 8^3\)
\(\ds \) \(=\) \(\ds 2 \times 2^3 + 4^3 + 3 \times 6^3 + 10^3\)
\(\ds \) \(=\) \(\ds 8 \times 6^3\)
\(\ds \) \(=\) \(\ds 2^3 + 2 \times 4^3 + 5 \times 6^3 + 8^3\)
\(\ds \) \(=\) \(\ds 5 \times 3^3 + 4 \times 6^3 + 9^3\)

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:

\(\ds 12^3\) \(=\) \(\ds 2^3 + 3^3 + 3 \times 4^3 + 5^3 + 4 \times 6^3 + 8^3\)
\(\ds \) \(=\) \(\ds 6 \times 3^3 + 4^3 + 5^3 + 3 \times 6^3 + 9^3\)
\(\ds \) \(:\) \(\ds \)
\(\ds \) \(=\) \(\ds 9 \times 3^3 + 4 \times 4^3 + 4 \times 5^3 + 9^3\)
\(\ds \) \(=\) \(\ds 2^3 + 5 \times 3^3 + 7 \times 4^3 + 5 \times 5^3 + 8^3\)

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.


Historical Note

This result can be found in Curious and Interesting Numbers by David Wells from $1986$, but it is a mystery as to why it was presented.

There appears to be absolutely nothing special about $351 \, 120$, as there are many cubes far smaller which possess the same property.

It is possible that Wells misinterpreted something that caught his eye, but googling for properties of $351 \, 120$ reveals that it is spectacularly uninteresting.


Sources