A Research On The Solutions To Equation Of Euler Function And Pseudo Smarandache Function

Euler Function φ（n）,Generalized Euler Function φ₂（n） and Pseudo SmarandacheFunction Z（n） play an important part in the field of number theory,and the research onthe solutions to functional equation of Numerial Theory is also regarded as a crucial topicof number theory.In this thesis,the above-mentioned three function equations and theircombinatorial equations are discussed by learning the basic knowledge of number theoryand refers to a large number of literature pairs.The major research findings are asfollows:In the first part,the problem of Positive integer solutions of equation involvingTernary function and Quaternary function is discussed,such as φ（abc）=φ（a）+3φ（b）+5φ（c）,φ（xyz）=φ（x）+φ（y）+φ（z）+6,φ（abcd）=φ（a）+φ（b）+2[φ（c）+φ（d）],Andall Positive integer solutions of the equation are given.In the second part,Based on the first part Euler function,the problem of Positive integer solution of equation involving Generalized Euler function φ（xyz）=φ₂ （X）+φ₂ （y） +φ₂（z） is discussed.And all Positive integer solutions （x,y,z）=（1,2,3）,（2,1,3）, （3,1,2）,（3,2,1）,（1,3,2）,（2,3,1）,（1,1,2）,（1,2,1）,（1,2,2）,（2,1,1）,（2,1,2）,（2,2,1） of the equa- tion are given.In the third part,The Solvability of composition equation of Z（n²）=φ₂（n²）, Z（n²）=φ₂（n）,and Z（n²）=φ₂2（n） are analyzed.The equation Z（n²）=φ₂（n）has only the n=1 Positive integer solution n=1.Equation Z（n²）=φ₂（n²） and Z（n²）=φ₂2（n） have no Positive integer solution.