On the Terai-Jésmanowicz Conjecture  

作　　者：Jian Ye XIA Ping Zhi YUAN 

机构地区：[1]Department of Basic Science, Guangdong University of Finance, Guangzhou 510521, P. R. China [2]School of Mathematics, South China Normal University Guangzhou 510631, P. R. China

出　　处：《Acta Mathematica Sinica,English Series》2008年第12期2061-2064,共4页数学学报（英文版）

基　　金：NSF of China (No.10571180);the Guangdong Provincial Natural Science Foundation (No.8151027501000114)

摘　　要：In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）. Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.

关 键 词：exponential diophantine equations Terai-Jesmanowicz conjecture Lucas sequences 

分 类 号：O156.4[理学—数学]