Hilbert-Waring Theorem/Particular Cases/4

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

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 $k = 4$ is:

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

That is:

$\map g 4 = 19$


Proof


Also see


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:

$g \left({4}\right) \ge 19$

Joseph Liouville subsequently showed that:

$g \left({4}\right) \le 53$

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

Thus:

$g \left({4}\right) = 19$

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


Sources