%0 Journal Article %T 丢番图方程(75n)^x+ (308n)^y= (317n)^z
On the Diophantine Equation (75n)^x+ (308n)^y= (317n)^z %A 黄日娣 %A 邓乃娟 %J Pure Mathematics %P 3358-3364 %@ 2160-7605 %D 2023 %I Hans Publishing %R 10.12677/PM.2023.1311348 %X 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). %K Jesmanowicz猜想，丢番图方程，正整数解
Jesmanowicz’s Conjecture %K Diophantine Equation %K Positive Integer Solution %U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=76431