Hilbert-Waring Theorem/Sequence

Sequence

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 integer sequence of values of $\map g k$ begins:

$1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055, \ldots$

This sequence is A002804 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Historical Note

The Hilbert-Waring Theorem was conjectured by Edward Waring in $1770$ in Meditationes Algebraicae, and was generally referred to as Waring's problem.

It was proved by David Hilbert in $1909$.

The assertion is that for each $k$ there exist such a number $\map g k$.

The problem remains to determine what that $\map g k$ actually is.