Smarandache Concerning The Nature And Function Of The Equation

It is well known that the Smarandache problems plays an important role in the study of number theory,because they related to many famous number theoretic problems. Therefore, Any fundamental progress in this field will con-tribute to the development of elementary number theory. American-Romanian famous number theorist Florentin Smarandache introduced hundreds of interest-ing sequences and number theory functions,and presented many problems and conjectures in his life. In1991, he published a book titled" Only problems,Not so-lutions", he presented105unsolved arithmetical problems and conjectures about these functions and sequences in it. Many researchers studied these sequences and functions from this book and obtained many important results.This dissertation mainly studied some aspects about Smarandache prob-lems.The main achievements contained in this dissertation are as follows.1. For any positive k≥2and any positive integer n, we call ak(n) as the Smarandache k-power complement member of n,if a,k(n) is the smallest positive integer such that nak(n) is a perfect k-th power number. That is ak(n)=min{u:u·n=mk;u,n∈N}. The k-power Smarandache Ceil func-tion is defined as Sk(n)=min{m:n|mk}. The main purpose of this paper is using the elementary method to give an equation involving of this two functions, then using the analytic method to get an asymptotic formula for the mean value of Sk(n).2. Discussing two special aspects congruence of Smarandache function.when2P-1is a prime or2P-1=P1P2…pr, we have S(2P-1)=1(mod p).3. Discussing the solution of Smarandache-totient function Zt(p)=p, the definition of Zt(n) is the minimum integer which can be divided by∑km=1φ(m), here the φ(n) is Eular function.