Several Types Of Equations Contain The Number Theoretic Function And Its Solution

Number theory has a long history,from Numbers start with some simple arithmetic problems, after thousands of years of development, the ancient sub-ject, charming, in the rapid development of science and technology still plays an indispensable role in modern times, some theory problems of research and has great and far-reaching significance. Along with some new problems have been proposed and attract a large number of scholars to study and solve it, and the problem that the equations have been an important content in the study of number theory.This paper mainly through the elementary method to study the Smaran-dache function; sum of divisors function and some other special arithmetic func-tion equation can solve problem of specific research results include the following aspects of content:1.For any positive integer n, the famous Smarandache function S(n) defined as the smallest positive integer m such that n|m!. That is, S(n)=min{m:m€N,n|m!}. The pseudo Smarandache function Z1(n) defined as the smallest integer m such that n|m(m+1)(2m+1)/6or Z1(n) min{m:m€N,n|12＋22＋…＋m2}, where N denotes the set of all posi-tive integers. The main purpose is to study the solvability of the equation Z1(n)＋1=S(n), and using the elementary method to give its all positive solu-tions, at the same time, we also give their exact representation of all solutions.2.For any positive integer n, let n=p1α1p2α2…Prαr denote the factorization of n, and let σ(n) denote the sum of divisors of n. In this paper, using some elementary number theory methods, the positive integer solutions (k,n) of the functional equation σ(n)=k(n+1) are discussed. We prove that if r=2, then the equation has only the positive integer solutions (k,n)=(2,2a1p2), where p2=2a1+1-3.3.Let b be a positive odd integer with b>3. In this paper, using some ele-mentary methods and the properties of congruence, the positive integer solutions (x, y, n) of the equation2yny-x=(b+2)x－bx are discussed, and all solutions of the equation are determined for b≠7(mod8), which part solved the problems of the solvability of the equation.