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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:
$$
\begin{cases}
x_1+x_2+\dots+x_n=N_1,\\
x_1^2+x_2^2+\dots+x_n^2=N_2.
\end{cases}
$$
The analysis is carried out as in the proof of a local limit theorem of probability theory and involves estimates of Weyl sums.
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
Linking options:
https://www.mathnet.ru/eng/mzm7620 https://www.mathnet.ru/eng/mzm/v18/i1/p19
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Abstract page: | 190 | Full-text PDF : | 71 | First page: | 1 |
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