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

The peculiarities of cellular automata visualization in nanoelectronics

“…Тот же способ применим и при свертке 2D-КА к 1D-модели визуализации. Его применение [Zenkin, 1990] к ряду натуральных чисел средствами системы PYTHAGORAS дало результаты в аддитивной теории чисел; идея состояла в том, чтобы выписать числа строчками по 16. Верно и обратное: небольшой по размеру поля 2D-КА можно развернуть в линию и отображать также, как 1D-КА.…”

The generalized waring problem: A new property of positive integers

“…Now, to prove PaWs theorem given in [1], it suffices to use the unique properties of the number 34 (instead of the number 169 used before). Here we present the shortest and simplest proof of Pall's theorem, since it is a good illustration of the method (see [3]) for proving general mathematical conjectures with the help of computers. An essential step in this proof is the computer verification of a number-theoretic predicate on an initial segment of positive integers [1, no], where the bound no is defined explicitly.…”

“…However, by using the cognitive computer graphics (CCG) dialogue system designed for number theory [3], it was shown that the number 169 is not an exception in this sense. For exarrlple, even among the numbers n < 100, the squares of the following positive integers The author and his colleagues were also the first who found that, among n < 100, the positive integers 34 45 50 53 61 65 68 72 73 74 82 89 90 97 98 can be written as sums of squares of positive integers for all s, 2 < s _~ n, except for the seven valucs s=n-z, zEZ(1,2).…”

“…In this case, expressing 34 as the sum of 6 -Sl squares, we see that each positive integer n > 34 can be written as the sum of exactly s = (6 -sl) + sl = 6 squares of positive integers. The direct computer verification (which is visual when using the CCG dialogue system for the number theory) of the finite segment [1,33] leads to the following mathematical fact: and this yields [3] IN(I, 2, 6)1 = Iz(1,2)l ; g(1,2) < 6. In the earlier paper [7], we obtained the following recurrent upper bound for the function G(m, r), provided that the value of g(rn -1, r) is known.…”

“…Consider the problem of representing positive integers n > 1 as sums of the form In the papers [1][2][3], by analogy with the classical Waring problem, the problem of proving that g(m, r) is finite for all m _> 1 and r > 2 is called the generalized Waring problem, because it is clear that for m = 0 we have In the simplest case m = 1 and r = 2, this theorem leads to a rather unexpected consequence. Note that here the set Z(1, 2) = {1,2, 4, 5, 7, 10, 13} is the famous Pall set discovered in 1933 during Pall's investigation of Waring's problem for sums of squares of positive integers [4][5][6].…”