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Chebyshevskii Sbornik, 2014, Volume 15, Issue 3, Pages 31–47 (Mi cheb351)  

This article is cited in 5 scientific papers (total in 5 papers)

Waring’s problem involving natural numbers of a special type

S. A. Gritsenkoab, N. N. Motkinac

a M. V. Lomonosov Moscow State University
b Financial University under the Government of the Russian Federation, Moscow
c Belgorod State University
Full-text PDF (621 kB) Citations (5)
References:
Abstract: In 2008–2011, we solved several well–known additive problems such that Ternary Goldbach's Problem, Hua Loo Keng's Problem, Lagrange's Problem with restriction on the set of variables. Asymptotic formulas were obtained for these problems. 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)=|m|<e2πim(ηN0,5k(a+b))sinkπm(ba)πkmk.
appear.
These series were investigated by the authors.
Suppose that k2 and n1 are naturals. Consider the equation
xn1+xn2++xnk=N(1)
in natural numbers x1,x2,,xk. The question on the number of solutions of the equation (1) is Waring's problem. Let η be the irrational algebraic number, n3,
kk0={2n+1,if 3n10,2[n2(2logn+loglogn+5)],if n>10.
In this report we represent the variant of Waring's Problem involving natural numbers such that a{ηxni}<b, where a and b are arbitrary real numbers of the interval [0,1).
Let J(N) be the number of solutions of (1) in natural numbers of a special type, and I(N) be the number of solutions of (1) in arbitrary natural numbers. Then the equality holds
J(N)I(N)σk(N,a,b).

The series σk(N,a,b) is presented in the main term of the asymptotic formula in this problem as well as in Goldbach's Problem, Hua Loo Keng's Problem.
Bibliography: 20 titles.
Keywords: Waring’s Problem, additive problems, numbers of a special type, number of solutions, asymptotic formula, quadratic irrationality, irrational algebraic number.
Received: 09.06.2014
Document Type: Article
UDC: 511.34
Language: Russian
Citation: S. A. Gritsenko, N. N. Motkina, “Waring’s problem involving natural numbers of a special type”, Chebyshevskii Sb., 15:3 (2014), 31–47
Citation in format AMSBIB
\Bibitem{GriMot14}
\by S.~A.~Gritsenko, N.~N.~Motkina
\paper Waring’s problem involving natural numbers of a special type
\jour Chebyshevskii Sb.
\yr 2014
\vol 15
\issue 3
\pages 31--47
\mathnet{http://mi.mathnet.ru/cheb351}
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  • https://www.mathnet.ru/eng/cheb351
  • https://www.mathnet.ru/eng/cheb/v15/i3/p31
  • This publication is cited in the following 5 articles:
    1. A. V. Shutov, “Ob odnoi additivnoi zadache, svyazannoi s razlozheniyami po lineinoi rekurrentoi posledovatelnosti”, Chebyshevskii sb., 24:3 (2023), 228–241  mathnet  crossref
    2. A. A. Zhukova, A. V. Shutov, “Additivnaya zadacha s k chislami spetsialnogo vida”, Materialy IV Mezhdunarodnoi nauchnoi konferentsii “Aktualnye problemy prikladnoi matematiki”. Kabardino-Balkarskaya respublika, Nalchik, Prielbruse, 22–26 maya 2018 g. Chast II, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 166, VINITI RAN, M., 2019, 10–21  mathnet  crossref
    3. A. A. Zhukova, A. V. Shutov, “Geometrizatsiya sistem schisleniya”, Chebyshevskii sb., 18:4 (2017), 222–245  mathnet  crossref  elib
    4. S. A. Gritsenko, N. N. Motkina, “O razreshimosti uravneniya Varinga v naturalnykh chislakh spetsialnogo vida”, Chebyshevskii sb., 17:1 (2016), 37–51  mathnet  elib
    5. A. A. Zhukova, A. V. Shutov, “Binarnaya additivnaya zadacha s chislami spetsialnogo vida”, Chebyshevskii sb., 16:3 (2015), 246–275  mathnet  elib
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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