数论函数方程Kφ2[X(X+1)/2=S(SL(x^(19)))]的可解性

The solvability of the arithmetic function equation Kφ_(2)[X(X+1)/2=S(SL(x^(19)))]

令φ_(2)(x)为广义欧拉函数,S(x)为Smarandache函数,SL(x)为Smarandache LCM函数,利用初等数论的方法及数论函数方程的性质,给出丢番图方程Kφ_(2)[X(X+1)/2=S(SL(x^(19)))]的全部解为:(k,x)=(22,2),(42,3),(20,4),(20,5),(7,6),(4,10),(5,15).

Suppose thatφ_(2)(x)is the generalized Euler function S(x),is the Smarandache function,and SL(x)is the Smarandache LCM function.By using the method of elementary number theory and the properties of function equation in number theory,all solutions of the number theory function equation Kφ_(2)[X(X+1)/2=S(SL(x^(19)))]are given as follows:(k,x)=(22,2),(42,3),(20,4),(20,5),(7,6),(4,10),(5,15).