On the Unboundedness of the Number of Natural Solutions for a Parameter-Dependent System of Equations

In this paper, we consider a system of 10 equations from the standpoint of the number of its natural solutions, containing a non-negative integer parameter n and describing the magic state of the corresponding special table of numbers. As a result of the study, it is constructively proven that, for each natural number m, there exist natural numbers nm and sm such that, for a non-negative integer parameter n equal to nm, this system has at least 2m solutions, and all ten coordinates of each of these solutions are sm-digit natural numbers, with the first, ninth, and tenth coordinates in decimal notation being represented only by the digits 0, 8, and 9, and the d-th coordinate, d∈{2,3,…,8}, being represented only by a single digit, equal to (d−1). This result, which constructively confirms the unboundedness of the number of solutions of this system depending on a non-negative integer parameter n, strengthens some recently published results.