数论函数方程mφ(n)=φ_(2)(n)+S(n^(10))的解

Positive Integer of Solutions to Ternary Euler Function Equation mφ(n) =φ_(2)(n)+S(n^(10))

利用Euler函数φ(n)、广义Euler函数φ_(2)(n)与Smarandache函数S(n)的性质,结合初等方法讨论了数论函数方程mφ(n)=φ_(2)(n)+S(n^(10))的可解性。证明了该方程只在m=1、2、4、6、8、11、13时才有正整数解,并给出了其全部正整数解。

This paper makes using of the properties of Euler functionφ(n),generalized Euler functionφ_(2)(n) and Smarandache function S(n),the solvability of number theory function equation mφ(n)=φ_(2)(n)+S(n^(10)) is discussed.It is proved that the equation has positive integer solutions only when m=1,2,4,6,8,11,13,and all positive integer solutions are given.