丢番图方程(an)^(x)+(bn)^(y)=(cn)^(z),a^(2)+b^(2)=c^(5)  

On the Diophantine Equation (an)^(x)+(bn)^(y)=(cn)^(z),a^(2)+b^(2)=c^(5)

作　　者：邓乃娟 DENG Nai-juan(Department of Mathematics,ZhanJiang Preschool Education College,Zhanjiang 524000,China)

机构地区：[1]湛江幼儿师范专科学校数学系,广东湛江524000

出　　处：《南宁师范大学学报（自然科学版）》2023年第3期10-14,共5页Journal of Nanning Normal University：Natural Science Edition

基　　金：湛江市非资助科技攻关计划项目(2021B0521)

摘　　要：设n,a=|m(m^(4)-10m^(2)+5)|,b=5m^(4)-10m^(2)+1,c=m^(2)+1是正整数,且2|m,m>0.该文证明了:若(x,y,z,n)≠(2,2,5,1),(2,2,3,c 2),(2,2,4,c)是丢番图方程(an)^(x)+(bn)^(y)=(cn)^(z)的正整数解,则x<y<z或y<x<z;当(a,b,c)=(38,41,5),(404,1121,17)时,丢番图方程(an)^(x)+(bn)^(y)=(cn)^(z)的正整数解只有(x,y,z,n)=(2,2,5,1),(2,2,3,c 2).Let n,a=|m(m^(4)-10m^(2)+5)|,b=5m^(4)-10m^(2)+1,c=m^(2)+1 be positive integers with 2|m,m>0.It is proved that,if(x,y,z,n)≠(2,2,5,1),(2,2,3,c 2),(2,2,4,c)is a positive integer solution of the Diophantine equation(an)^(x)+(bn)y=(cn)^(z),then x<y<z or y<x<z.It is also shown that,when(a,b,c)=(38,41,5),(404,1121,17),the Diophantine equation(an)^(x)+(bn)y=(cn)^(z) has only positive integer solutions(x,y,z,n)=(2,2,5,1),(2,2,3,c 2).

关 键 词：JESMANOWICZ猜想 丢番图方程 正整数解 

分 类 号：O156[理学—数学]