Research On Problems Including Euler Function And Smarandache Function

Euler function and Smarandache function are one of the objects that need to be paid attention to in the learning process of number theory function.Previously,many scholars at home and abroad have studied these two kinds of number theory functions in depth and promoted the development of number theory.In this thesis,we use elementary methods,analytical methods and classified discussion ideas to discuss the problems involving Euler function and Smarandache function.The main contents are as follows:In the first part,we discuss the problem of positive integer solutions of two equations containing Euler functions.One is to study the solvability of the equationφ(abcd)= φ(a)+2φ(b)+3φ(c)+4φ(d),from which all the positive integer solutions of the equation are given.The other is to study the solvability of the equation containing composite functionsφ(φ2(x-φ2(x)))=2,so as to all the positive integer solutions of the equation.In the second part,the solvability of an equation including Euler function and Smarandache function S(SL(n))=φ2(n)is analyzed,and all positive integer solutions are given.In the third part,the,β-th hybrid mean of the difference between Smarandache multiplier function SM(n)and maximum prime factor function P(n)is studied,and the asymptotic formula of the mean distribution is obtained,and the complete proof process is given.