Hilbert-Waring Theorem/Particular Cases/3

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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:

$\map g 3 = 9$


Proof


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)