广义Ferm at商中的平方数和立方数

李江华

(西安理工大学理学院,陕西 西安 710048)

广义Ferm at商中的平方数和立方数

李江华

(西安理工大学理学院,陕西 西安 710048)

设p是奇素数,a和b是适合a＞b,gcd(a,b)=1以及pab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.

广义Fermat商;平方数;立方数;三项Diophantine方程

1 引言及结论

2 若干引理

3 定理1.1的证明

4 定理1.2的证明

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The squares and cubes in generalized Ferm at quotients

Li Jianghua

(College of Science,X i′an University of Technology,X i′an 710048,China)

Let p be an odd prim e,and let a,b be positive integers such that a＞b,gcd(a,b)=1 and pab.In this paper we discussed the generalized Ferm at quotient problem s under these conditions.Using the elem entary method and some recent results on ternary Diophantine equations.Proved that if p＞13,then(ap−1−bp−1)/p is not a square,and if p＞7,then it is not an odd cube.It hasm ade som e p rogress for the generalized Ferm at quotient prob lem s.

generalized Fermat quotient,square,cube,ternary Diophantine equation

O156.4

A

1008-5513(2012)06-0774-05

2012-05-21.

陕西省自然科学基金(2012K 06-43);陕西省教育厅专项计划基金(12JK 0874).

李江华(1980-),博士,研究方向:解析数论及其应用.

2010 M SC:11D 61