Computers as novel mathematical reality: V. Easier Waring problem
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.
References
N. A. Vavilov, “Computers as novel mathematical reality. I. Personal Account,” Computer tools in education, no. 2, pp. 5–26, 2020 (in Russian); doi: 10.32603/2071-2340-2020-2-5–26
N. A. Vavilov, “Computers as novel mathematical reality. II. Waring Problem,” Computer tools in education, no. 3, pp. 5–55, 2020 (in Russian); doi: 10.32603/2071-2340-2020-3-5-55
N. A. Vavilov, “Computers as novel mathematical reality. III. Mersenne numbers and divisor sums,” Computer tools in education, no. 4, pp. 5–58, 2020 (in Russian); doi: 10.32603/2071-2340-2020-4-5-58
N. A. Vavilov, “Computers as novel mathematical reality. IV. Goldbach Problem” Computer tools in education, no. 4, pp. 5–71, 2021 (in Russian); doi: 10.32603/2071-2340-2021-4-5-71
N. A. Vavilov, V. G. Khalin, and A. V. Yurkov, Mathematica dlya nematematika [Mathematica for nonmathematician], Moscow: MCCME, 2021 (in Russian).
A. S. Verebrussov, “Sur l’equation x 3 + y 3 + z 3 = 2u 3 ,” Mat. Sb., vol. 26, no. 4, pp. 622–624, 1908 (in Russian).
S. V. Volkov, Generalitet Rossiiskoi Imperii: entsiklopedicheskii slovar’ generalov i admiralov ot Petra I do Nikolaya II., vol. 1, vol. 2., Moscow: Tsentropoligraf, 2009 (in Russian).
S. S. Demidov, “K istorii teorii lineinykh differentsial’nykh uravnenii,” Istoriko-matematicheskie Issledovaniya,
vol. 28, pp. 78–98, 1985 (in Russian).
V. A. Demjanenko, “On equation x 3 + y 3 + z 3 − t 3 = 1,” Izv. Vyssh. Uchebn. Zaved. Mat., no. 5, p. 57, 1965 (in
Russian).
This work is licensed under a Creative Commons Attribution 4.0 International License.
