# Hilbert-Waring Theorem/Particular Cases/3

## Particular Case of the Hilbert-Waring Theorem: $k = 3$

The Hilbert-Waring Theorem states that:

For each $k \in \Z: k \ge 2$, there exists a positive integer $\map g k$ such that every positive integer can be expressed as a sum of at most $\map g k$ $k$th powers.

The case where $k = 3$ is:

Every positive integer can be expressed as the sum of at most $9$ positive cubes.

That is:

$\map g 3 = 9$

## 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.

## Sources

• 1909: Arthur WieferichBeweis des Satzes, daß sich eine jede ganze Zahl als Summe von höchstens neun positiven Kuben darstellen läßt (Math. Ann. Vol. 66: 95 – 101)