About The Equation Of The Smarandache Function And Gaussian Function And Nature

The properties of prime number is the kernel of the research in number theory. Many researchers had studied and explored them deeply, and obtained a lot of very meaningful results. French mathematician Fermat conjectured that all the numbers such as Fn= 22n+1 are prime numbers, we call these numbers as Fermat number. It caused scholars to research about the properties of Fermat number. Furthermore,the research about the properties on the number theory function is the core of number theory.Based on the above reasons on Fermat number and Smarandache function, we use elementary and analytic methods to study the function S(Fn), and an-alyze the solvability of three equations about Gauss function. The dissertation gets the following results:1.A new function related to Smarandache function S(Fn) is defined and its lower bound is studied by using the elementary and analytic methods.2.Using the properties of Gauss function [x], the solvability of three equa-tions such as are discussed and its partial solutions are obtained.3.The solvability of equation S(Z(n)= Z(S(n)) is discussed, and the solu-tion of four conditions such as n= p, n= 2p, n = 3p and n = p1P2···pk,2< Pi< P2<···< Pk are gotten.