欧拉方程φ(abcd)=φ(a)+2φ(b)+3φ(c)+4φ(d)+6的正整数解

Positive Integer Solution to Euler Functionφ(abcd)=φ(a)+2φ(b)+3φ(c)+4φ(d)+6

φ(n)定义为Euler函数,研究了一个常数为特殊完全数的四元不定方程φ(abcd)=φ(a)+2φ(b)+3φ(c)+4φ(d)+6的可解性的问题.利用初等方法与技巧,对主要结论进行了完整的阐明,得到了该方程的所有正整数解.文章所用的分类方法以及将常数选定为特殊的完全数的思想,为同类型方程的研究提供了新的思路.

φ(n)defined as Euler function,the solvability of a quaternion indefinite equationφ(abcd)=φ(a)+2φ(b)+3φ(c)+4φ(d)+6 whose constant is a special perfect number is studied.By using elementary methods and techniques,the main conclusions are expounded completely,and all positive integer solutions of the equation are obtained.The classification method used in this paper and the idea of choosing constants as special perfect numbers provide a new idea for the study of the same type equations.