Computers As a Novel Mathematical Reality: II. Waring's Problem

In this part I discuss the role of computers in the current research on additive number theory, in particular, in the solution of the classical Waring problem. In its original 18th century form, this problem consisted in finding for each natural k the smallest s= g(k) such that all natural numbers can be written as sums of s nonnegative kth powers: n = x(1)(k) +...+ x(s)(k) In the 19th century, the problem was modified as the quest of finding such a minimal s = G(k) that almost all can be expressed in this form. In the 20th century, this problem was further specified as finding such and the precise list of exceptions. The 20th century problem is still unsolved even for cubes. However, even the solution of the original Waring problem was [almost] finalized only in 1984, with heavy use of computers. In the present paper, we document the history of this classical problem and its solution, discuss the possibilities of using this and surrounding material in education, and cover some further related aspects.