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.
Read more about this topic: Schnirelmann Density
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