Equation Involving The Smarandache Function And Euler Function And Its Nature

As we all know, one of the main elements of number theory is to study the properties of arithmetical functions. Lots of mathematicians studied the properties of all kinds of arithmetical functions and got many valuable conclu-sions about them, which promoted a further development of number theory. The book "Only problems, Not solutions!" published in 1991 attracted many people's attention, which was written by Romanian expert of number theory Florentin Smarandache. In this book, professor F.Smarandache put forward 105 arithmetical problems and conjectures about special sequences and arithmetical functions. Many scholars researched and explored these problems deeply and got many valuable conclusions. Furthermore, some researchers proposed and studied some new and interesting problems about Smarandache functions and the classical arithmetic functions.Based on the interest in Smarandache functions, the dissertation makes researches on the solvability of some equations about Smarandache functions by using elementary and analytic ways. Specifically, the dissertation gets the following results:1. The solvability of a congruent equation about is studied, and all the prime number solutions of the equation are given by using the elementary methods.2. The equation including Sk(n) and Euler function is studied by the method of classification, and all the positive integer solutions are obtained.3. According to the properties of Z(n) and Z*(n), the equation Z(n)+ Z*(n) = n is discussed, and its even solution is gotten when n belongs to n= 2k and n=2p1p2…pk,2