On the Analysis of a System of Equations Containing a Parameter n and Describing a Special State of a Certain Table of Numbers

In this paper, we study a system of 10 equations containing a parameter—a non-negative integer n—and associated with the problem of filling a special table with natural numbers. As a result, by developing a hybrid approach—first, by proving an a priori estimate for the solution of the system, which implies a finite set of solutions, thereby significantly narrowing the search space for a solution, and second, by performing a computer calculation of all remaining vectors satisfying the a priori estimate—we established that the system is not solvable for all values of the parameter n. We also prove a criterion for $(x_0,x_1,\dots ,x_9)\in {\mathbb{N}}^{10}$ to be a solution to the system for some value of the parameter n. Furthermore, we prove a fact that, in particular, implies that the set of values of the parameter n for which the system has at least 10 solutions is countable.