华东师范大学学报(自然科学版) ›› 2009, Vol. 2009 ›› Issue (5): 138-141.

• 应用数学与基础数学 • 上一篇    

几个不定方程在mathbf{Q}(\sqrt{-3})中的解（英）

王永亮   

  1. 菏泽学院 数学系, 菏泽 274015
  • 收稿日期:2008-09-24 修回日期:2009-02-10 出版日期:2009-09-25 发布日期:2014-10-13
  • 通讯作者: 王永亮

Solutions to some Diophantine equations over mathbf{Q}(\sqrt{-3}

WANG Yong-liang   

  1. Department of Mathematics, Heze University, Heze, 274015, China
  • Received:2008-09-24 Revised:2009-02-10 Online:2009-09-25 Published:2014-10-13
  • Contact: WANG Yong-liang

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摘要： 应用Fermat下降法，证明了不定方程~{x^{4}-y^{4}=z^{2}} 与~{x^{4}+4y^{4}=z^{2}} 在 {\mathbf{Q}}(\sqrt{-3}) 没有非平凡解, 它表明Fermat方程当~n=4时在此域中仍然没有非平凡解.

关键词: Fermat下降法, 代数整数环, 虚二次域, Fermat下降法, 代数整数环, 虚二次域

Abstract: By using Fermat’s method of descent, this paper proved that Diophantine equations { x^{4}-y^{4}=z^{2}} and { x^{4}+4y^{4}=z^{2}} have no non-trivial solutions over {\mathbf{Q}}(\sqrt{-3}), which implies that the Fermat Equation also has no non-trivial solutions in this field for n =4.

Key words: \quad ring of algebraic integers, \quad imaginary quadratic fields, Fermat’s method of descent, \quad ring of algebraic integers, \quad imaginary quadratic fields

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