Discussion Of Two Types Of Indefinite Equation Solution And The Study Of Mean Value Of Smarandache Function

It is indefinite equation(called Diophantine equation)and Smarandache function's mean value issues that are considered as two crucial and more active research fields in Number Theory.They have extremely rich contents,however,there are still some unsolved problems which inspire the research interests of many experts and scholars.In this paper,we mainly use elementary methods and analytic methods to study the solvability of two kinds of indefinite equations and the mean value problems of Smarandache function,the main results are as follows:Firstly,we discuss the problem of integer solutions of the indefinite equationx~3±a~3=Dy~2(D?0),by using the elementary methods such as recursive sequence,some properties of the Legendre symbol,congruent and the solutions to Pell equation.It is proved that the indefinite equationx~3+27=37y~2 only has the integer solution(x,y)=(-3,0).The indefinite equationx~3-27=37y~2 only has the integer solutions(x,y)=(3,0),(30,±27),(4,±1).And the problem of integer solutions of the indefinite equationx~3±1331=2pqy~2.Secondly,by using the elementary methods,we discuss integer solution problem of indefinite equation (na)~x+(nb)~y=(nc)~z.And we prove the indefinite equation (na)~x+(nb)~y=(nc)~z has no solution in positive integers other than(x,y,z)=(2,2,2)when a=20,b=99,c=101.Thirdly,the analytic methods is utilized to study the mean value distribution problem of Smarandache Ceil function and product of prime divisor function U(n),and given an interesting asymptotic formula.Finally,problem about mean value for Smarandache power functions SP(n)ismainly examined by elementary methods and analytic methods.In other words,the hybrid mean value between Smarandache power function SP(n)and arithmetical function R(n)is determined based on the sequence of simple number.