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 Mat. Zametki: Year: Volume: Issue: Page: Find

 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

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}