关于Diophantine方程(36n)^(x)+(323n)^(y)=(325n)^(z)的整数解  

Integer Solutions for Diophantine Equation(36n)^(x)+(323n)^(y)=(325n)^(z)

作　　者：常青[1] 高丽[1] CHANG Qing;GAO Li(School of Mathematics and Computer Science,Yan’an University,Yan’an 716000,China)

机构地区：[1]延安大学数学与计算机科学学院,陕西延安716000

出　　处：《延安大学学报（自然科学版）》2021年第3期54-56,60,共4页Journal of Yan'an University：Natural Science Edition

基　　金：国家自然科学基金资助项目(11471007);陕西省科技厅科学技术研究发展计划资助项目(2013JQ1019);延安大学科研计划资助项目(YD2014-05)。

摘　　要：Jesmanowicz猜想Diophantine方程(na)^(x)+(nb)^(y)=(nc)^(z)仅有正整数解(x,y,z)=(2,2,2),其中a,b,c是两两互素的正整数且满足a^(2)+b^(2)=c^(2)。主要运用简单同余法、奇偶分析法、二次剩余理论以及分类讨论等初等方法,证明了对任意的正整数n,Diophantine方程(36n)^(x)+(323n)^(y)=(325n)^(z)仅有正整数解(x,y,z)=(2,2,2)。即证明了:当(a,b,c)=(36,323,325)时Jesmanowicz猜想成立。The Jesmanowicz conjecture Diophantine equation has only positive integer solutions(x,y,z)=(2,2,2)when a,b and c are positive integers of reciprocal prime and satisfy a^(2)+b^(2)=c^(2).Through elementary methods such as congruence method,odd-even analysis,quadratic residual theory and classification,it was proved that the Diophantine equation had only positive integer solutions(x,y,z)=(2,2,2)for any positive integer.The Jesmanowicz conjecture was true if(a,b,c)=(36,323,325).

关 键 词：JESMANOWICZ猜想 DIOPHANTINE方程 正整数解 初等方法 

分 类 号：O156[理学—数学]