Some Of The Smarandache Function Equation Solvability Problem,

The equation involving Smarandache functions is one of the most important topics which has attracted the attention of many domestic and foreign number-theoretic experts and scholars recently in number theory research.Based on the interests in the equation involving Smarandache function, this paper mainly put research on the definitions and properties of pseudo Smaran-dache function Z(n)and its dual function Z*(n), F.Smarandache LCM function SL(n)and its dual function SL*(n), the Smarandache reciprocal function Sc(n), and explore the solvability of a series special equations including the functions above. It was proved that the equation has infinite positive integer solutions, and confirmed all concrete forms of every solution by using the elementary and combinatorial method and discuss separately, especially via congruent equation. specifically, the main achievements in this paper include the following:1. The equations involving two types of Smarandache functions:Z(n)+SL(n)=n; Z(n)+Z*(n)=n; Z(n)+SL*(n)=n; Z(n)=SL*(n); Z(n)=SL*(n)+1; SL*(n)=Z,(n); SL(n)+SL*(n)=n; Sc(n)=Z*(n)+n; Sc(n)=SL(n); Sc(n)+SL(n)=2n; Sc(n)=SL(n)+1; Sc(n)=SL*(n).All concrete forms of solutions of the 12 equations above are achieved, only to find that equation Sc(n)=SL(n) and Sc(n)+SL(n)=2n has the same solutions related to Fermat prime.2. The solvability of the equations involving three types of Smarandache functions is discussed, and all the positive integer solutions for equation Sc(n)= SL(n)+SL*(n) are achieved.3. Discuss the solvability of the equations involving four types of Smaran-dache functions, get some of the positive integer solutions for equation Z(n)+ Z*(n)=SL(n)+SL*(n).4.The solutions of equations Sc(n)=SL*(n)+n and Sc(n)+Z(n)=2n will finally turn into the discussion of prime.