Diophantione方程x^(d(n))+y^(d(n))=z^(φ(n))的本原解

Primitive Solutions of the Diophantine Euqation x^(d(n))+y^(d(n))=z^(φ(n))

对于正整数n,设d(n)和φ(n)分别是除数的函数和Euler函数,又设p是奇素数.证明了:当n=1,2,4或p时,方程xd(n)+yd(n)=zφ(n)有无穷多组本原解(x,y,z);当n≠1,2,4,p或p2时,该方程无本原解(x,y,z).

Let n be a positive integer,and let d（ n ） and φ（ n ） denote the divisor function and Euler＇ s totient function respectively .Let p be an odd prime. It is proved that if n = 1,2,4,or p ,then the equation x^d（n）＋y^d（n）=zφ（n） has infinitely many primitive solutions （x, y, z）;if n ≠ 1,2,4 p or p^2, then the equation has no primitive solution （x, y,z）.