ON THE SOLVABILITY OF WARING’S EQUATION INVOLVING NATURAL NUMBERS OF A SPECIAL TYPE
https://doi.org/10.22405/2226-8383-2016-17-1-37-51
Abstract
This paper is a continuation of our research on additive problems of number theory with variables that belong to some special set. We have solved several well–known additive problems such that Ternary Goldbach’s Problem, Hua Loo Keng’s Problem, Lagrange’s Problem,Waring’s Problem. Asymptotic formulas were obtained for these problems with restriction on the set of variables. The main terms of our formulas differ from ones of the corresponding classical problems. In the main terms the series of the form
σk(N, a, b) =X |m|<∞ e2πim(ηN−0,5k(a+b)) sink πm(b − a) πkmk. appear. These series were investigated by the authors. Let η be the irrational algebraic number, a and b are arbitrary real numbers of the interval [0, 1]. There are natural numbers x1, x2, . . . , xk such that a 6 {ηxni } < b. In this paper we evaluate the smallest k for which the equation xn1 + xn2 + . . . + xnk = N is solvable.
About the Authors
S. A. Gritsenko
МГУ имени М. В. Ломоносова, Финансовый университет при Правительстве РФ
Московский государственный технический университет имени Н. Э. Баумана
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor of the chair Mathematics of the 1 Financial University under the Government of the Russian Federation, professor of the mechanics and mathematics faculty of the M. V. Lomonosov Moscow State University
N. N. Motkina
Белгородский государственный национальный исследовательский университет
Russian Federation
Russian Federation
Candidate of Physico-Mathematical Sciences, Associate Professor, Associate Professor at the Department of Mathematics, Faculty of Mathematics and Science Education, Institute of Pedagogy, Belgorod National Research University
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Review
For citations:
Gritsenko S.A., Motkina N.N. ON THE SOLVABILITY OF WARING’S EQUATION INVOLVING NATURAL NUMBERS OF A SPECIAL TYPE. Chebyshevskii Sbornik. 2016;17(1):37-51. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-1-37-51
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