Euler商中的p次方幂

The p-th Power in Euler Quotients

对于正整数n,设φ(n)和ω(n)分别是n的Euler函数和n的不同素因子的个数.对于适合a>1以及gcd(a,n)=1的正整数a,形如(aφ(n)-1)/n的正整数称为Euler商.设p是奇素数,根据高次Diophantine方程的性质讨论了Euler商中p次方幂.证明了:当ω(n)≥3时,Euler商都不是p次方幂.

For any positive integer n,Letφ（n）andω（n）denote the Euler function and the number of distinct prime divisors of nrespectively.For any positive integer asuch that a〉1 and gcd（a,n）=1,the positive integer of the form（aφ（n）-1）/nis called an Euler quotient.Let pbe an odd prime.In this paper,by the properties of higher degree diophantine equation,the p-th powers in Euler quotients are discussed.The result（aφ（n）-1）/nis not a p-th powers is proved ifω（n）≥3.