Hilbert-Waring Theorem/Particular Cases/3

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

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 $\map g k$ positive $k$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

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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

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

1912: Aubrey Kempner: Bemerkungen zum Waringschen Problem (Math. Ann. Vol. 72: pp. 387 – 399)

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Waring's problem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Waring's problem
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Waring's problem