Hilbert-Waring Theorem/Particular Cases/4

Particular Case of the Hilbert-Waring Theorem: $k = 4$

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 $k = 4$ is:

Every positive integer can be expressed as the sum of at most $19$ powers of $4$.

That is:

$\map g 4 = 19$

Proof

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Also see

Positive Integers not Expressible as Sum of Fewer than 19 Fourth Powers

Historical Note

It is clear that some integers require at least $19$ powers of $4$ to represent them as a sum, as $79$ requires $19$ of them:

Thus Edward Waring knew that:

$\map g 4 \ge 19$

Joseph Liouville subsequently showed that:

$\map g 4 \le 53$

Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most $19$ fourth powers.

Finally it was shown in $1986$ by Ramachandran Balasubramanian, Jean-Marc Deshouillers and François Dress that $\map g 4 \le 19$.

Thus:

$\map g 4 = 19$

and so Waring's claim has been shown to be true.

Sources

1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $19$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $19$
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