丢番图方程(75n)^x+ (308n)^y= (317n)^z  

On the Diophantine Equation (75n)x+ (308n)y= (317n)z

作　　者：黄日娣 邓乃娟 

机构地区：[1]湛江幼儿师范专科学校数学系,广东 湛江

出　　处：《理论数学》2023年第11期3358-3364,共7页Pure Mathematics

摘　　要：设a,b,c是两两互素的正整数且a2+b2=c2。Jesmanowicz猜想:对于任意给定的正整数n,方程(an)x+(bn)y=(cn)z只有正整数解(x,y,z)=(2,2,2)。本文利用数论中的一些方法证明了:对任意的正整数n,方程(75n)x+ (308n)y= (317n)z只有正整数解(x,y,z)=(2,2,2),即当(a,b,c)=(75,308,317)时,Jesmanowicz猜想成立。Let a,b,c be a primitive Pythagogrean triples such that a2+b2=c2. Jesmanowicz conjectured that, for any positive integer n, the Diophantine equation (an)x+(bn)y=(cn)z has only positive integer solution (x,y,z)=(2,2,2). In this paper, by using some methods of number theory,we prove that, for any positive integer n, the Diophantine equation (75n)x+ (308n)y= (317n)z has only positive integer solution (x,y,z)=(2,2,2), that is the Jesmanowicz conjecture is true, when (a,b,c)=(75,308,317).

关 键 词：JESMANOWICZ猜想 丢番图方程 正整数解 

分 类 号：O15[理学—数学]