On the Equation x = count(d, x) + n and Its Application

In this paper, we consider a discrete equation in natural numbers of the form x=count(d,x)+n, where n is a natural number, and count(d,x) is the number of occurrences of the digit d∈{0,1,…,9} in the decimal representation of the number x, and estimate the cardinality of the set of its solutions. As a result of the study, it is proved that for each natural number n and each digit d∈{2,3,…,9}, the number of natural solutions of this discrete equation does not exceed two. It is established that in the case d=1, for each natural number n, the number of natural solutions of this discrete equation does not exceed three. It is constructively proved by developing an appropriate algorithm that in the case d=0, for each natural number k there exists a natural number nk, such that the number of different natural solutions to the equation x=count(0,x)+nk, even written in decimal notation using no more than three digits such as 0, 8, 9, and having the same number of digits, is not less than k. We also demonstrate the application of the results and of the technique developed and presented in this paper to a system of equations describing the magic state of a special table of numbers, which strengthens and complements some recently obtained results.