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Mat. Zametki, 1975, Volume 18, Issue 1, Pages 19–25 (Mi mz7620)  

A two-dimensional additive problem with an increasing number of terms

Sh. A. Ismatullaev

Mathematical Institute, Academy of Sciences of UzSSR

Abstract: In this paper there is established an asymptotic formula for the number of simultaneous representations of two numbers as sums of an increasing number of terms involving a power function, i.e., an asymptotic (as $n\to\infty$) formula is found for the number of solutions in integers $x_i$, $0\le x_i\le p$, of the following system of diophantine equations:
$$ \begin{cases} x_1+x_2+…+x_n=N_1,
x_1^2+x_2^2+…+x_n^2=N_2. \end{cases} $$
The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.

Full text: PDF file (464 kB)

English version:
Mathematical Notes, 1975, 18:1, 592–596

Bibliographic databases:

UDC: 511.2
Received: 09.07.1973

Citation: Sh. A. Ismatullaev, “A two-dimensional additive problem with an increasing number of terms”, Mat. Zametki, 18:1 (1975), 19–25; Math. Notes, 18:1 (1975), 592–596

Citation in format AMSBIB
\Bibitem{Ism75}
\by Sh.~A.~Ismatullaev
\paper A~two-dimensional additive problem with an increasing number of terms
\jour Mat. Zametki
\yr 1975
\vol 18
\issue 1
\pages 19--25
\mathnet{http://mi.mathnet.ru/mz7620}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=382163}
\zmath{https://zbmath.org/?q=an:0325.60023}
\transl
\jour Math. Notes
\yr 1975
\vol 18
\issue 1
\pages 592--596
\crossref{https://doi.org/10.1007/BF01461136}


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