Several Contain Equations Of The Smarandache Function

Number Theory,among the mathematical disciplines,occupies an important position.The famous Smarandache problems play an important role in the study of number theory,it was first introduced by American-Romanian number theorist Florentin Smarandache.In 1993,he published a book entitled "Only problems,Not solutions".In this book,he presented 105 unsolved arithmetical problems and conjectures about these functions and sequences.The Smarandache's universe is undoubtedly very fascinating.Even though sometime this universe has a very simple structure from number theory standpoint,it doesn't cease to be deeply mysterious and interesting.Following the Smarandache spirit,Felice Russo presents some new Smarandache functions and solved and unsolved problems in the book of "A set of new Smarandache functions,sequences and conjectures in number theory".In this dissertation,we studied some problems of this book and gave the answer.The main achievements contained in this dissertation are as follows:1.On the basis of the Smarandache function S(n)and Euler functionφ(n), we studied the equationφ(n)=S(n~2)andφ(n)=S(n~3) and solved them by using the elementary methods.2.On the basis of the Pseudo-Smarandache-Squarefree function Zw(n)and Pseudo-Smarandache function Z(n),we studied some equations and solved them completely by using the elementary methods.