Hilbert-Waring Theorem/Variant Form/Particular Cases

Particular Cases of the Hilbert-Waring Theorem -- Variant Form

For each $k \in \Z: k \ge 2$, there exists a positive integer $G \left({k}\right)$ such that every sufficiently large positive integer can be expressed as a sum of at most $G \left({k}\right)$ $k$th powers.

The case where $k = 2$ is proved by Lagrange's Four Square Theorem‎:

$G \left({2}\right) = 4$

That is, every sufficiently large positive integer can be expressed as the sum of at most $4$ squares.

The case where $k = 3$ is:

Every sufficiently large positive integer can be expressed as the sum of a number of positive cubes.

The exact number is the subject of ongoing research, but at the time of writing ($11$th February $2017$) it is known that it is between $4$ and $7$.

That is:

$4 \le \map G 3 \le 7$

The case where $k = 4$ is:

Every sufficiently large positive integer can be expressed as the sum of at most $16$ $4$th powers.

That is:

$\map G 4 = 16$

The case where $k = 7$ is:

The exact number is the subject of ongoing research, but at the time of writing ($20$th December $2018$) it is known that it is between $8$ and $33$.

That is:

$8 \le \map G 3 \le 33$