Waring's problem from the standpoint of the cognitive interactive computer graphics

Abstract

It is well known that an apt drawing is sometimes capable of generating new ideas in a human mind. The modern Interactive Computer Graphics (ICG) allows the effective use of the cognitive graphics function (in contrast to the more traditional and common illustrative one) even in the most abstract fields of science. A “knowledge-generating” man-machine ICG system, DSNT, — The Dialogue System for ICG-investigations in the additive Number Theory—has been worked out on the basis of the cognitive ICG concept. In the framework of the ICG-system, DSNT, an original method has been worked out for the dynamic visualization of abstract number-theoretic objects in the form of some colour-musical ICG-images or the so-called pythograms of these objects. Under certain conditions these pythograms can create some new mathematical ideas and hypotheses in an investigator's imagination and provide the ways of their strict proofs. This paper describes some new results in the well-known field of the abstract number theory-classical Waring's problem. These results have been obtained by means of cognitive ICG. The author also attempts to describe, or, to put it more precisely, to picture how these results and their proofs are created. The general philosophy of the ICG-visualization of scientific abstractions briefly discussed in the paper can be useful for direct investigations of intuitive creative meta-procedures of our thinking. These investigations will undoubtedly become an important trend in the field of artificial intelligence and computer techniques for self-knowledge and self-education