Hilbert-Waring Theorem

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

The assertion is that for each $k$ such a number $g \left({k}\right)$ exists, but then you have the problem of finding that $g \left({k}\right)$.

k = 2
From Lagrange's Four Square Theorem‎ we have $g \left({2}\right) = 4$.

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

k = 3
Waring knew that $g \left({3}\right) \ge 9$ as:

In fact these are the only two integers that need as many as $9$ cubes to express them.

All other integers need no more than $8$.

It has recently been shown that only finitely many numbers do require $8$ cubes. From some point on, $7$ cubes are enough.

k = 4
It is clear that $g \left({4}\right) \ge 19$, as $79$ requires $19$ fourth-powers.

Waring claimed that $g \left({4}\right) = 19$, and this has in fact recently been shown to be true.

Proof
Waring conjectured it in 1770.

It was proved by David Hilbert in 1909 and hence this result is sometimes known as the Hilbert-Waring Theorem.

Lagrange's Four Square Theorem‎ of 1770 shows that $g \left({2}\right) = 4$. This had first been conjectured by Bachet in 1621. Fermat claimed to have found a proof, but if so he never published it.

Liouville 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.