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

## 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.