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

D. A. Mit'kin

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 339-346

Résumé

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.