广义Fermat商中的平方数和立方数

The squares and cubes in generalized Fermat quotients

设p是奇素数,a和b是适合a>b,gcd(a,b)=1以及pab的正整数.在这些条件下讨论了一类广义Fermat商为完全平方及完全立方问题.利用初等方法以及三项Diophantine方程的最新结果,证明了当p>13时,(ap-1 bp-1)/p不是平方数;当p>7时,(ap-1 bp-1)/p不是奇立方数.对广义Fermat商的方幂问题做出了实质性进展.

Let p be an odd prime, and let a, b be positive integers such that a 〉 b, gcd（a, b） = 1 and p＇ ab. In this paper we discussed the generalized Fermat quotient problems under these conditions. Using the elementary method and some recent results on ternary Diophantine equations. Proved that if p 〉 13, then （ap-1 - bV-1）/p is not a square, and if p 〉 71 then it is not an odd cube. It has made some progress for the generalized Fermat quotient problems.