数论函数方程kφ(n)=7φ_(2)(n)+S(n^(13))的正整数解

The Positive Integer Solutions of Arithmetic Function Equation kφ(n)=7φ_(2)(n)+S(n^(13))

对于正整数n,数论函数φ(n),φe(n)和S(n)分别为Euler函数、广义Euler函数和Smarandache函数.讨论了包含φ(n),φe(n)和S(n)三个数论函数的方程kφ(n)=7φ_(2)(n)+S(n13)的可解性,基于这三个数论函数方程的性质,用初等方法证明了该数论函数方程只在k=1,4,5,6,7,8,9,11,14,17,23时有正整数解,并给出了具体的正整数解.

For positive integer n,the arithmetic functionφ(n)is the Euler function,the arithmetic functionφ_(2)(n)is generalized Euler function and arithmetic function S(n)is the Smarandache function.The solvability of the arithmetic function equation kφ(n)=7φ_(2)(n)+S(n^(13))involving three arithmetic functionsφ(n),φe(n)and S(n)was discussed.Based on the properties of these three arithmetic functions,we obtained the equation has positive integer solutions only when k=1,4,5,6,7,8,9,11,14,17,23,and gave its specific positive integer solu-tions by using the elementary method.