丢番图方程x^(2)+(2n)^(2)=y^(9)(1≤n≤7)的整数解  

The Integer Solution of the Diophantine Equations x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z,1≤n≤7)

作　　者：陈一维 柴向阳 CHEN Yi-wei;CHAI Xiang-yang(College of Mathematics and Statistics,North China University of Water Resources and Electric Power,Zhengzhou 450045,China)

机构地区：[1]华北水利水电大学数学与统计学院,郑州450045

出　　处：《重庆工商大学学报（自然科学版）》2021年第1期92-98,共7页Journal of Chongqing Technology and Business University:Natural Science Edition

摘　　要：在高斯整环中,利用代数数论理论和同余理论的方法研究丢番图方程x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z,1≤n≤7)的整数解问题;首先统计了1≤n≤7时已有的证明结果,之后在n=3,5,6,7时对x分奇数和偶数情况讨论,证明了n=3,5,6,7时丢番图方程x^(2)+(2n)^(2)=y^(9)无整数解,即证明了丢番图方程x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z,1≤n≤7)无整数解。In Gauss domain,the problem of integer solution of the Diophantine equation x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z,1≤n≤7)is discussed by using the methods of algebraic number theory and congruence theory.First of all,finding out the results that have been proven when 1≤n≤7.Then,by discussing the two cases that x is odd and x is even respectively,we proved that the Diophantine equation x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z)has no integer solution when n=3,5,6,7.Finally the conclusion is reached that the Diophantine equation x^(2)+(2n)^(2)=y^(9)(x,y,n∈Z)has no integer solution when 1≤n≤7.

关 键 词：高斯整环 代数数论 同余理论 丢番图方程 整数解 

分 类 号：O156.4[理学—基础数学]