关于本原商高数的Miyazaki猜想  

The Miyazaki Conjecture on Primitive Pythagorean Numbers

作　　者：王枭涵 苟素[2] 

机构地区：[1]西安外国语大学经济金融学院,陕西西安710127 [2]西安邮电大学理学院,陕西西安710121

出　　处：《数学的实践与认识》2014年第8期287-290,共4页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11371291);陕西省教育厅项目(12JK0883)

摘　　要：设a,b,c是适合a=2^(2r)-n^2,b=2^(r+1)n,c=2^(2r)+n^2的正整数,其中r是正整数,n是奇素数.运用初等数论方法讨论了指数Diophantine方程c^x+b^y=a^z.证明了:当2~r=n+1时,方程仅有正整数解(x,y,z)=(1,1,2);否则,方程无解。上述结果部分地证实了有关本原商高数的Miyazaki猜想。Let a, b, c be positive integers such that a = 22r - n2, b = 2r＋1n, c = 22r ＋ nz where r is a positive integer, n is an odd prime. In this paper, using some elementary number theory methods, the exponential diophantine equation cx ＋ by = az is discussed. We prove that if 2r = n ＋ 1, then the equation has only the positive integer solution （x,y, z） -= （1, 1, 2）, otherwise, it has no solution. This result partly Pythagorean numbers. verifies Miyazaki＇s conjecture on primitive

关 键 词：本原商高数 指数DIOPHANTINE方程 Miyazaki猜想 

分 类 号：O156[理学—数学]