与多个数论函数相关的复合函数方程解的存在性

Existence of Solutions for Complex Function Equations Related to Several Number Theory Functions

对一个由Smarandache函数S(n)、伪Smarandache函数Z(n)、另一个F.Smarandache可乘函数S(n)和简数根函数sim(n)复合的数论函数方程Z(S(n))=S(sim(n))解的存在性进行研究.利用初等和解析等方法与技巧,首先借助简数根函数的定义和特性进行系统的分类,避免可能解出现重叠或遗漏现象,再结合Excel软件中的简单函数编程对所有可能的解进行数据筛选,最终发现该方程具有有限个解,并通过多种表达方式对方程所有解进行了表示.同时,在研究过程中,为后续选题、研究同一类由多个数论函数复合的方程,以及出现存在多个解的情况并对其进行表达方式简化等相关问题提供了很好的思路,并给出了关于简数根函数的几个新的运算性质以及一些简便地计算方法.

The existence of a solution of a number-theoretic function equation Z(S(n))=S(sim(n))composed of Smarandache function S(n),apseudo-Smarandache function Z(n),F.Smarandache multiplier function S(n)and simplified root function sim(n)is studied.In this paper,the methods and techniques of elementary and resolution are used to classify the system by means of the definition and characteristics of the simple root number functions,so as to avoid possible overlap or omission,and then combine the simple root function programming in the Excel Software to filter all possible solutions.The equation is finally found to have a finite solution and all solutions of the equation are solved by a variety of expressions a representation is made.At the same time,in the course of the study,it provides a good idea for the follow-up topics,the study of a class of equations which are compounded by a number of numbers of functions,and the problems of the existence of multiple solutions and the simplification of the expression.In this paper,several new computing properties and some simple calculation methods for the number of simple root functions are given.