Some Number Theoretic Function, Equation Solving

As we all know, Number theory is one of the oldest knowledge branch. but the most profound secrets and extraordinary truth are closely linked. The main content that elementary number theory contains is to study the Property of number-theoretic function. The well-known F.Smarandache function S(n) is one of the important arithmetical functions. as well as Professor F.Smarandache have presented 105 unsolved arithmetical problems and conjectures about special sequences and arithmetical functions. The unsolved problem and conjectures aroused many scholars' interest. With the posing of the problem, many scholars obtained some important and theoretical valued results on the theory. The research results on square complement number and the dual function is very significant!In this dissertation, we study he equation solutions related to number-theoretic function and F.Smarandache unsolved problem. At the end we have given all the solutions about equations and some asymptotic formula. more specifically, The main achievements embodied in this dissertation are as follows:1. The F.Smarandache square complement number have many significant character. The many equations related to square complement number play an crucial link in number-theoretic study. this dissertation use the distinguished results of Goldbach Conjecture to study the solvability of the equations involving square complement number. at the same time, this dissertation prove the equation have infinite positive integer solutions.2. The F.Smarandache dual function S~*(n) is a very important multiplicative function. The properties of The F.Smarandache dual function S~*(n) are very similar to The Smarandache S(n). 3. The Smarandache S(n) occupies an important position in the study of the number-theoretic function. In this dissertation, we use elementary methods and analytic methods to study the solvability of two equations and give give its all positive integer solutions about the equations.4. with regard to the Smarandache LCM function SL(n), this dissertation use elementary methods to study the mean value property of SL(n), and obtain a sharper asymptotic formula for it. Simultaneously, we study the extremum of the two limits.