Euler方程φ(xy)=k_1φ(x)+k_2φ(y)(k_1≠k_2)的正整数解

The Integer Solutions of Euler Equation φ(xy) =k_1φ( x) + k_2φ( y)( k_1≠k_2)

讨论了一个形如φ(xy)=k_1φ(x)+k_2φ(y)(k_1≠k_2)的具体方程φ(xy)=5φ(x)+7φ(y)的可解性,给出了其一切整数解.并根据这一方程的解的情况,给出了(x,y)=(k_1+k_2,k_1+k_2)是方程φ(xy)=k_1φ(x)+k_2φ(y)(k_1≠k_2)的1组整数解的结论,这里的k_1,k_2都是正整数.

The solvability of a specific equation φ（ xy） = 5φ（ x） ＋ 7φ（ y）,such as φ（ xy） = k1φ（ x） ＋k2φ（ y）,was discussed. And all integer solutions were given. According to the condition of its solutions,a conclusion that（ x,y） =（ k1＋ k2,k1＋ k2） was a positive integer solution of equation φ（ xy） = k1φ（ x） ＋k2φ（ y） was given,where k1≠k2,and k1,k2 were positived integers.