Waring's Problem

Given a positive integer k, we ask the following question. Is there a positive integer s such that every positive integer can be represented as a sum of s perfect k-th powers of nonnegative integers? For instance, take the case when k=2. A theorem of Lagrange says that every positive integer is the sum of four squares. As examples:

$$3= 1^2+1^2+1^2+0^2 \hbox{ and } 23= 3^2+3^2+2^2+1^2.$$

In the case k=3, the fewest number of cubes need to add up to 23 is 9, as follows:

23=2³+2³+1³+1³+1³+1³+1³+1³+1³.

It turns out that all positive integers may be represented as a sum of nine or fewer cubes. In fact, only one other integer besides 23 requires as many as nine. In the late nineteenth century, Hilbert showed that the answer was always yes to the above question. Only recently, the smallest possible value of s has been completely determined for all k.