| 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x2+(2n)2=y9（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x2+(2n)2=y9无整数解,即证明了丢番图方程x2+(2n)2=y9（x,y,n∈〖WTHZ〗Z〖WTBX〗,1≤n≤7)无整数解。 |
| 关键词: 高斯整环 代数数论 同余理论 丢番图方程 整数解 |
| DOI： |
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| 基金项目: |
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| The Integer Solution of the Diophantine Equations x2+(2n)2=y9（x,y,n∈〖WTHZ〗Z〖WTBX〗,1≤n≤7) |
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CHEN Yi-wei, CHAI Xiang-yang
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College of Mathematics and Statistics,North China University of Water Resources and Electric Power, Zhengzhou 450045, China
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| Abstract: |
| In Gauss domain,the problem of integer solution of the Diophantine equation x2+(2n)2=y9（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 x2+(2n)2=y9（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 x2+(2n)2=y9（x,y,n∈〖WTHZ〗Z〖WTBX〗) has no integer solution when 1≤n≤7. |
| Key words: Gauss integral ring algebraic number theory congruence theory Diophantine equation integer solution |
