陈一维, 柴向阳.丢番图方程x2+(2n)2=y9(1≤n≤7)的整数解[J].重庆工商大学学报(自然科学版),2021,38(1):92-98
CHEN Yi-wei, CHAI Xiang-yang.The Integer Solution of the Diophantine Equations x2+(2n)2=y9(x,y,n∈〖WTHZ〗Z〖WTBX〗,1≤n≤7)[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2021,38(1):92-98
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∈〖WTHZ〗Z〖WTBX〗,1≤n≤7)
  
DOI:
中文关键词:  高斯整环  代数数论  同余理论  丢番图方程  整数解
英文关键词:Gauss integral ring  algebraic number theory  congruence theory  Diophantine equation  integer solution
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作者单位
陈一维, 柴向阳 华北水利水电大学 数学与统计学院郑州 450045 
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中文摘要:
      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∈〖WTHZ〗Z〖WTBX〗,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∈〖WTHZ〗Z〖WTBX〗,1≤n≤7)无整数解。
英文摘要:
      In Gauss domain,the problem of integer solution of the Diophantine equation x2+(2n)2=y9(x,y,n∈〖WTHZ〗Z〖WTBX〗,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∈〖WTHZ〗Z〖WTBX〗) 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∈〖WTHZ〗Z〖WTBX〗) has no integer solution when 1≤n≤7.
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