Abstract: 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 ｘｄ（ｎ）＋ｙｄ（ｎ）＝ｚφ（ｎ） has infinitely many primitive solutions （ｘ，ｙ，ｚ）;if　ｎ≠１，2，４　ｐ  or ｐ2，then the equation has no primitive solution　（ｘ，ｙ，ｚ）.

Key words: higher Diophantine equation；primitive solution；divisor function；Euler&rsquo, s totient function