This is a very short introduction to Waring's problem, designed for non-mathematicians. Clicking on a link should give you a very short definition of that term, or a link to a biography of that mathematician. There is a slightly longer version of this introduction in which I have tried to put Waring's problem in a mathematical context. You can find it here. (I hope that making it longer will actually make it easier to understand!)
Waring's problem is about the possibility of writing positive whole numbers as sums of kth powers (e.g. squares or cubes). Here's a precise statement of it.
Waring's problem: Take any whole number k that is greater than or equal to 2. Show that there is some number s (that is allowed to depend on k) so that every positive whole number is a sum of s kth powers.
This was first solved by Hilbert, and then Hardy and Littlewood gave another proof a few years later. For example, it is known that every positive whole number is a sum of four squares, so when k = 2 we may take s = 4. (This is actually Lagrange's theorem, about which I have written a little here.) It is also known that every positive whole number is a sum of nine cubes, so when k = 3 we may take s = 9. Note, however, that to solve Waring's problem as I've stated it, it is not necessary to give an explicit s, only to show that one exists. Finding the smallest s that works is much harder than just finding some value that works (although it is definitely an interesting problem).
Hardy and Littlewood used their circle method. They wrote down an expression (an integral, in fact) for the number of ways to write N as a sum of s kth powers. Since this is a number of ways to do something, it must be 0 or a positive whole number. They then estimated this quantity (which is not trivial to do, and also was perhaps not an obvious thing to try), and showed that they could find an s so that if N is big enough, then the number of ways to write N as a sum of s kth powers is positive. Since this must also be a whole number, it must be at least 1 — so there must be some way to write N as a sum of s kth powers.
kth powers: 1k = 1, 2k, 3k, 4k, …, and we'll include 0k = 0 too.
Square numbers: 12 = 1 × 1 = 1, 22 = 2 × 2 = 4, 32 = 3 × 3 = 9, 42 = 4 × 4 = 16, …. Here, we'll also include 02 = 0 × 0 = 0.
Cubes: 13 = 1 × 1 × 1 = 1, 23 = 2 × 2 × 2 = 8, 33 = 3 × 3 × 3 = 27, 43 = 4 × 4 × 4 = 64, …. Again, we'll include 03 = 0 × 0 × 0 = 0.
A sum of s kth powers: For example, 10 is a sum of four squares: 10 = 9 + 1 + 0 + 0 = 32 + 12 + 02 + 02.
This page last updated on 18th January 2009.