Cubes which are Sum of Three Cubes
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
- Feb. 1937: A. Russell and C. E. Gwyther: The Partition of Cubes (Math. Gazette Vol. 21, no. 242: pp. 33 – 35) www.jstor.org/stable/3605742
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $216$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $729$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $216$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $729$