# Hilbert-Waring Theorem/Particular Cases/3/Historical Note

## 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:
 $\displaystyle 23$ $=$ $\displaystyle 2^3 + 2^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3 + 1^3$ $\displaystyle 239$ $=$ $\displaystyle 4^3 + 4^3 + 3^3 + 3^3 + 3^3 + 3^3 + 1^3 + 1^3 + 1^3$
The fact that $\map g 3 = 9$ was established from $1909$ to $1912$ by Arthur Josef Alwin Wieferich and Aubrey John Kempner.