Hilbert-Waring Theorem/Particular Cases/3/Historical Note
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Particular Case of the Hilbert-Waring Theorem: $k = 3$: Historical Note
Edward Waring knew that some integers required at least $9$ positive cubes to represent them as a sum:
| \(\ds 23\) | \(=\) | \(\ds 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3\) | ||||||||||||
| \(\ds 239\) | \(=\) | \(\ds 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3\) |
The fact that $\map g 3 = 9$ was mostly established by Arthur Josef Alwin Wieferich $1909$.
The remaining gap in the argument was resolved in $1912$ by Aubrey John Kempner.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem