丢番图方程(75n)^x+ (308n)^y= (317n)^z

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

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。

Abstract: 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).