An estimate for the number of terms in the Hilbert–Kamke problem. II
Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 339-346
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It is proved that there exist integers $A_1,\dots,A_n$ such that the system of congruences $$ \sum^s_{i=1}\binom{x_i}j=A_j(\bmod 2^{\alpha(n,j)}),\qquad j=1,\dots,n, $$ 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 $x_1,\dots,x_s$ 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.
@article{SM_1988_60_2_a4,
author = {D. A. Mit'kin},
title = {An~estimate for the number of~terms in~the {Hilbert{\textendash}Kamke} {problem.~II}},
journal = {Sbornik. Mathematics},
pages = {339--346},
year = {1988},
volume = {60},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1988_60_2_a4/}
}
D. A. Mit'kin. An estimate for the number of terms in the Hilbert–Kamke problem. II. Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 339-346. http://geodesic.mathdoc.fr/item/SM_1988_60_2_a4/
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[3] Arkhipov G. I., “O probleme Gilberta–Kamke”, Izv. AN SSSR. Ser. matem., 48:1 (1984), 3–52 | MR | Zbl
[4] Gelfond A. O., Ischislenie konechnykh raznostei, Nauka, M., 1967 | MR