Компьютер как новая реальность математики: V. Легкая проблема Варинга

Abstract

In this part I continue the discussion of the role of computers in the current research on the additive number theory, in particular in the solution of the easier Waring problem. This problem consists in finding for each natural k the smallest such s =v(k) that all natural numbers n can be written as sums of s integer k-th powers n = ± x1k ± ... ± xsk with signs. This problem turned out to be much harder than the original Waring problem. It is intimately related with many other problems of arithmetic and diophantine geometry. In this part I discuss various aspects of this problem, and several further related problems, such as the rational Waring problem, and Waring problems for finite fields, other number rings, and polynomials, with special emphasys on connection with polynomial identities and the role of computers in their solution. As of today, these problems are quite far from being fully solved, and provide extremely broad terrain both for the use in education, and amateur computer assisted exploration.