Related Issues About Diophantine Equation Andpseudo Smarandache Function Research

With the continuous development of number theory research, the various forms of number theory problems are appear. Number Theory numerous unresolved problems attracting number theory expert and enthusiasts to research in number theory. In this paper, we use elementary method and analytic method to study two types of the solvability problem of Diophantine equation, and the equations, the mean and the lower bound estimation problem of Pseudo Smarandache function Z(n), and some results are given. The main results of this paper are as follows:1. By using the elementary method, we studied the solvability problem of the Diophantine equation 3 2x ?1 ?301y and given the proof. In the same way, we proved the Je?manowícz’s conjecture is established when a ?44, b ?117, c ?125. That is when a ?44, b ?117, c ?125, the Diophantine equation()()()x y zna ?nb ?nc has only integeral solution(x, y, z) ?(2, 2, 2).2. By using the elementary method and analytic method, we studied some problems about the Pseudo Smarandache function. First of all, we study the solvability of two equations()()k?n ?Z n and Z(n) ??(n) ?2n, and by using elementary methods and the properties of the Pseudo Smarandache function and the Euler function. All positive integer solutions of them are given. The second, we study the mean value properties of the composite function Sdf(Z(n)) about the Smarandache double factorial function and the Pseudo Smarandache function, and give a sharper asymptotic formula. The last, we study a lower bound estimate problem of the Pseudo Smarandache function at the special value 2 1p?. The sharper lower bounds of the Pseudo Smarandache function is given.