The Research On The Related Problems About Euler Function Equation And Smarandache Function

In the number theory,the problem of the solution of Diophantus equation and the mean value of the function plays a very powerful role.A majority of scholars at home and abroad have carried out in-depth research on it and achieved many remarkable achievements.Based on this,the properties of Euler function and Smarandache function are studied further by using the related theories and methods of elementary number theory and analytic number theory in this master’s dissertation.Specifically,the main contents of this master’s dissertation include the following aspects:1.Using elementary method,the author discusses the solvability problem of the Euler function equationφ(xy)=m(φ(x)+ φ(y))when nm = 6,3k and φ(xyz)-m(φ(x)+ φ(y)|φ(z))when m=10,and obtains their all positive integer solutions.2.Using the method of classification discussion,the author studies the solvability problem of an equation of including the Euler function φ(n)and the Smarandache double factorial function Sdf(n)Sdf(n)= 2φ(n)and gives the all positive integer solutions of this equation.3.Using the analytical method,the author studies the hybrid mean value of Smarandache double factorial function Sdf(n)and the prime factor number functionΩ(n),and obtains the following asymptotic formula:(?)where αi,{i=1,2,…,k)are computable constants.