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

The mean problem of Diophantine equation and Smarandache function is two extremely important topics in number theory.Their research results greatly enrich the content of number theory,but there are still some unsolved problems that have attacted the attention and research of many scholars.In this master’s thesis,we use the elementary method,analytical method and algebraic munber theory to study the problem of solving two kinds of Diophantine equations and the mean value problem of Smarandache function.The main results are as follows:1.The problen of integer solutions of Diophantine equationAx~2+B=Cy~n is discussed by means of algebraic number theory,prove that the Diophantine equation x~2+1024=y~1 ~1 have only integer solution(x,y)=(±32,2),when n=7,the Diophantine equation x~2+256=4y~n have only integer solution(x,y)=(±16,2),when n=11,13,15,the Diophantine equation x~2+256=4y~n have no integer solution.2.The problem of positive integer solutions of the Diophantine equation?(mn)=a?(m)+b?(n)+c is discussed by elementary method,prove that the Diophantine equation?(mn)=a?(m)+b?(n)+c have respectively 27 and 20 solutions when(a,b)=(3,4),(5,6),c=16.When a~2+b~2=c~2,gcd(a,b,c)=1,the Diophantine equation have no integer solution,the case when c is the perfect number and c=ab.3.The elementary method and analytic method were performed to study the Smarandache LCM function SL(n)and Smarandache function S(n)and the mean value problem ofS_δ(n)(SL(n)-S(n))~2and a sharper asymptotic formula was proposed.