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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 3, Pages 345–351 (Mi msb1858)  

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

An estimate for the number of terms in the Hilbert–Kamke problem. II

D. A. Mit'kin


Abstract: It is proved that there exist integers $A_1,…,A_n$ such that the system of congruences
$$ \sum^s_{i=1}\binom{x_i}j=A_j(\bmod 2^{\alpha(n,j)}),\qquad j=1,…,n, $$
where $\alpha(n,j)$ denotes the exponent of the highest power of 2 dividing $(n!/(j-1)!)2^{[(n-j+1)/2]+1}$, is solvable in integers $x_1,…,x_s$ only if the necessary condition $s\geqslant H(n)$ holds, where
$$ H(n)=\sum_{0\leqslant k\leqslant[\ln n/\ln 2]}2^k(2^{[n/2^k]}-1). $$
From this the estimate $r(n)\geqslant H(n)$ is derived for the number $r(n)$ of terms in the Hilbert–Kamke problem. Combined with a result from the previous paper, this gives the formula $r(n)=H(n)$ for $n\geqslant12$.
Bibliography: 4 titles.

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English version:
Mathematics of the USSR-Sbornik, 1988, 60:2, 339–346

Bibliographic databases:

UDC: 511
MSC: Primary 11P05, 11D41, 11D72; Secondary 11L40, 11D85, 11P55
Received: 25.11.1985

Citation: D. A. Mit'kin, “An estimate for the number of terms in the Hilbert–Kamke problem. II”, Mat. Sb. (N.S.), 132(174):3 (1987), 345–351; Math. USSR-Sb., 60:2 (1988), 339–346

Citation in format AMSBIB
\Bibitem{Mit87}
\by D.~A.~Mit'kin
\paper An~estimate for the number of~terms in~the Hilbert--Kamke problem.~II
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 3
\pages 345--351
\mathnet{http://mi.mathnet.ru/msb1858}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=889596}
\zmath{https://zbmath.org/?q=an:0662.10013|0619.10013}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 339--346
\crossref{https://doi.org/10.1070/SM1988v060n02ABEH003172}


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    This publication is cited in the following articles:
    1. Trevor Wooley, “Vinogradov's mean value theorem via efficient congruencing”, Ann. Math, 175:3 (2012), 1575  crossref
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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