On the number of natural solutions of a specific discrete equation and its properties

The article studies a discrete equation in natural numbers of the form x = count (d, x) + n, where n is a natural number; count(d, x) is the number of occurrences of the digit d ∈ {0, 1, …, 9} in the decimal notation of the number x. The objective of the work is to obtain a priori estimates of solutions to the equation and to analyze the dependence of the number of solutions on the parameters d and n. The main result of the study is the establishment of a two-sided a priori estimate for solutions to the equation: n ≤ x ≤ n + ⌊lgn⌋ + 2. It follows from this estimate that for fixed d and n, the number of natural solutions is limited; the equation is not solvable for all values of n. For digits d ∈ {9, 8, …, 2} and any natural n, it is proved that the number of solutions does not exceed 2. For d = 1 and any natural n, it is shown that the number of solutions does not exceed 3. For d = 0, it is constructively substantiated that with increasing n, the number of natural solutions, each of which can be represented in decimal notation using only four digits, can increase indefinitely. Theoretical and practical significance. The results are of a theoretical-mathematical nature and can be applied in the study of similar discrete equations in natural numbers; in the analysis of mathematical puzzle games described by similar equations; they also complement known results in the field of Diophantine equations.

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