Waring’s problem

Waring asked whether it is possible to represent every natural number as a sum of bounded (http://planetmath.org/BoundedInterval) number of nonnegative $k$’th powers, that is, whether the set $\{n^k\mid n\in {\mathbb{Z}}_{+}\}$ is an additive basis (http://planetmath.org/Basis2). He was led to this conjecture by Lagrange’s theorem (http://planetmath.org/LagrangesFourSquareTheorem) which asserted that every natural number can be represented as a sum of four squares.

Hilbert [1] was the first to prove the conjecture for all $k$. In his paper he did not give an explicit bound on $g(k)$, the number of powers needed, but later it was proved that

$$g(k)=2^k+\lfloor {\left(\frac{3}{2}\right)}^k\rfloor -2$$

except possibly finitely many exceptional $k$, none of which are known.

Wooley[4], improving the result of Vinogradov[3], proved that the number of $k$’th powers needed to represent all sufficiently large integers is

$$G(k)\le k(\ln k+\ln \ln k+O(1)).$$