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Mat. Sb. (N.S.), 1970, Volume 82(124), Number 3(7), Pages 360–370 (Mi msb3455)  

This article is cited in 1 scientific paper (total in 1 paper)

An asymptotic formula for the number of solutions of a Diophantine equation

M. I. Israilov


Abstract: Suppose that $k$, $s$, $m_1,…,m_k$, $m_1',…,m_s'$ are fixed positive integers, $m$ is a fixed integer, $p$ is an increasing positive integer, and suppose that a sequence of integers $\{n_k\}$ satisfies the following conditions: 1) $n_{k+1}\geqslant n_k(1+k^{-1/2+\varepsilon})$ , where $\varepsilon>0$ is arbitrarily small; 2) for fixed $m,n,a,B$, the number of solutions of the Diophantine equation
$$ mn_{x+a}-nn_x=B $$
in $x$ in the half-open interval $[0,p)$ does not exceed some constant $q$ which does not depend on $m,n,a,B$.
Under these assumptions, an asymptotic formula with remainder term is derived for the number of solutions of the Diophantine equation
$$ m_1n_{x_1}+…+m_kn_{x_k}=m_1'n_{y_1}+…+m_s'n_{y_s}+m $$
in integers $0\leqslant x_1,…,x_k$; $y_1,…,y_s<p$.
The results obtained extend and refine several results obtained by other authors.
Bibliography: 7 titles.

Full text: PDF file (838 kB)
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English version:
Mathematics of the USSR-Sbornik, 1970, 11:3, 327–338

Bibliographic databases:

UDC: 511.222
MSC: 11D75, 26B35, 11Bxx
Received: 30.06.1969

Citation: M. I. Israilov, “An asymptotic formula for the number of solutions of a Diophantine equation”, Mat. Sb. (N.S.), 82(124):3(7) (1970), 360–370; Math. USSR-Sb., 11:3 (1970), 327–338

Citation in format AMSBIB
\Bibitem{Isr70}
\by M.~I.~Israilov
\paper An~asymptotic formula for the number of solutions of a~Diophantine equation
\jour Mat. Sb. (N.S.)
\yr 1970
\vol 82(124)
\issue 3(7)
\pages 360--370
\mathnet{http://mi.mathnet.ru/msb3455}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=323714}
\zmath{https://zbmath.org/?q=an:0211.37503|0215.34603}
\transl
\jour Math. USSR-Sb.
\yr 1970
\vol 11
\issue 3
\pages 327--338
\crossref{https://doi.org/10.1070/SM1970v011n03ABEH002072}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Zeév Rudnick, Alexandru Zaharescu, “The distribution of spacings between fractional parts of lacunary sequences”, form, 14:5 (2002), 691  crossref  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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