The Research Of Three Types Of Indefinite Equations Containing Number Theory Functions

Function of number theory plays an important role in the development and research of number theory.Euler function,Smarandache function and Smarandache LCM function are one of the key research objects.Many scholars at home and abroad have conducted in-depth research on these functions,which has promoted the development of number theory.In this paper,we will use the elementary method,analytic method and the idea of classification discussion to study the related problems involving these three funtions.The main content is divided into the following three aspects:In the first part,we discuss the solvable property of the complex function equationφ₂（φ（n-φ（φ（n））））=,12and give respective 13 and 15 positive integer solutions of corres-ponding equations.In the second part,we study the solvable property of the equation of shapeφ（abcd）=k₁φ（a）+k₂φ（b）+k ₃φ（c）+k₄φ（d）±C is studied,and the positive integer solutions of the corresponding three equations are given.In the third part,the solvable problem of equations related to Euler function,Smarandache function and Smarandache LCM function is studied,and positive integer solutions of the corresponding equations are obtained.