# Hilbert-Waring Theorem/Particular Cases/4/Historical Note

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

## Source of Name

This entry was named for David Hilbert and Edward Waring.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $9$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9$