Schnirelmann Density - Waring's Problem

Waring's Problem

Let and be natural numbers. Let . Define to be the number of non-negative integral solutions to the equation

and to be the number of non-negative integral solutions to the inequality

in the variables, respectively. Thus . We have

The volume of the -dimensional body defined by, is bounded by the volume of the hypercube of size, hence . The hard part is to show that this bound still works on the average, i.e.,

Lemma. (Linnik) For all there exists and a constant, depending only on, such that for all ,

for all

With this at hand, the following theorem can be elegantly proved.

Theorem. For all there exists for which .

We have thus established the general solution to Waring's Problem:

Corollary. (Hilbert 1909) For all there exists, depending only on, such that every positive integer can be expressed as the sum of at most many -th powers.