dsaitЦифровые решения и технологии искусственного интеллектаDigital Solutions and Artificial Intelligence Technologies3033-7097Финансовый университет при Правительстве Российской Федерации10.26794/3030-7097-2026-2-3-51-57dsait-75Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ, ЧИСЛЕНЫЕ МЕТОДЫ И КОМПЛЕКСЫ ПРОГРАММMATHEMATICAL MODELING, NUMERICAL METHODS AND SOFTWARE PACKAGESО количестве натуральных решений специфического дискретного уравнения и его свойствахOn the number of natural solutions of a specific discrete equation and its propertieshttps://orcid.org/0000-0003-3729-6143БаротовР. Н.BarotovR. N.

Рузибой Нумонжонович Баротов — преподаватель кафедры математического анализа им. профессора А. Мухсинова факультета математики

Худжанд

Ruziboy N. Barotov — Lecturer, Department of Mathematical Analysis named after Professor A. Mukhsinov, Faculty of Mathematics

Khujand

ruzmet.tj@mail.ruhttps://orcid.org/0000-0001-5047-7710БаротовД. Н.BarotovD. N.

Достонжон Нумонжонович Баротов — старший преподаватель кафедры математики и анализа данных факультета информационных технологий и анализа больших данных

Москва

Dostonjon N. Barotov — Senior Lecturer, Department of Mathematics and Data Analysis, Faculty of Information Technology and Big Data Analysis

Moscow

DNBarotov@fa.ruХуджандский государственный университет им. акад. Б. ГафуроваТаджикистанKhujand State University named after academician Bobojon GafurovTajikistanФинансовый университет при Правительстве Российской ФедерацииРоссияFinancial University under the Government of the Russian FederationRussian Federation202607092026235157Copyright © Баротов Р.Н., Баротов Д.Н., 20262026Баротов Р.Н., Баротов Д.Н.Barotov R.N., Barotov D.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.digitarin.ru/jour/article/view/75

В статье исследуется дискретное уравнение в натуральных числах вида x = count (d, x) + n, где: n — натуральное число; count (d, x) — количество вхождений цифры d ∈{0, 1, …, 9} в десятичную запись числа x.

Цель работы — получить априорные оценки решений уравнения и проанализировать зависимость числа решений от параметров d и n.

Основной результат исследования — установлена двусторонняя априорная оценка для решений уравнения: n ≤ x ≤ n + ⌊lgn⌋ + 2. Из нее следует: при фиксированных d и n число натуральных решений ограничено; уравнение разрешимо не для всех значений n. Для цифр d ∈{9, 8, …, 2} и любого натурального n доказано, что число решений не превышает 2. Для d = 1 и любого натурального n показано, что число решений не превышает 3. Для d = 0 конструктивно обосновано, что с ростом n число натуральных решений, каждое из которых может быть представлено в десятичной записи с использованием всего четырех цифр, может неограниченно возрастать.

Теоретическая и практическая значимость. Результаты имеют теоретико‑математический характер и могут быть применены: при исследовании аналогичных дискретных уравнений в натуральных числах; в анализе математических игр‑головоломок, описываемых подобными уравнениями; как дополнение к известным результатам в области диофантовых уравнений.

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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The authors declare that there are no conflicts of interest present.