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 required at least $19$ powers of $4$ to represent them as a sum, as $79$ requires $19$ of them:

Thus knew that:
 * $g \left({4}\right) \ge 19$

subsequently showed that:
 * $g \left({4}\right) \le 53$

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

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

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