廣義Ferm at商中的平方數(shù)和立方數(shù)

李江華

(西安理工大學理學院,陜西 西安 710048)

廣義Ferm at商中的平方數(shù)和立方數(shù)

李江華

(西安理工大學理學院,陜西 西安 710048)

設p是奇素數(shù),a和b是適合a＞b,gcd(a,b)=1以及pab的正整數(shù).在這些條件下討論了一類廣義Fermat商為完全平方及完全立方問題.利用初等方法以及三項Diophantine方程的最新結果,證明了當p＞13時,(ap-1?bp-1)/p不是平方數(shù);當p＞7時,(ap-1?bp-1)/p不是奇立方數(shù).對廣義Fermat商的方冪問題做出了實質性進展.

廣義Fermat商;平方數(shù);立方數(shù);三項Diophantine方程

1 引言及結論

2 若干引理

3 定理1.1的證明

4 定理1.2的證明

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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.

陜西省自然科學基金(2012K 06-43);陜西省教育廳專項計劃基金(12JK 0874).

李江華(1980-),博士,研究方向:解析數(shù)論及其應用.

2010 M SC:11D 61