【作者】 邬毅；

【导师】 罗明；

【作者基本信息】 重庆师范大学 ， 基础数学， 2007， 硕士

【摘要】 不定方程不仅自身发展异常活跃,而且全面应用于离散数学的其他各个领域,它对人们学习研究和解决实际问题有着重要的作用。因此,国内外有诸多学者对不定方程进行着广泛而深入的研究。随着不定方程的推动,代数数论取得了最初的形成和发展。目前,代数数论已成为数学中一个内容异常丰富的分支,也是研究不定方程的重要工具。作为代数数论中的重要组成部分,二次域及二次域中的算术对研究不定方程有着重要的作用。对某些D,其二次域Q ( D )是Euclid域,因此在这些二次代数整数环Q ( D )中算术基本定理成立,我们可以建立和Z中同样的整除理论去解某些不定方程。关于用代数数论知识解答不定方程x~2 + D = 4 pn( D>0)已经有不少研究工作,特别是通过讨论虚二次域类数的可除性来研究不定方程x~2 + D =4 pn(D是适合p D的正奇数,且D≠4 p r? 1,其中r是正整数)的解数。本文主要利用二次域中的重要理论、二次代数整数环Q ( D )中算术基本定理以及Maple程序,主要证明了六个不定方程x~2 + D = 4y~5(D=7,11,-5,-13,-21,-29)的所有整数解,分四个章节来说明。第一章综述了关于不定方程x~2 + D =4 pn及用代数数论解某些不定方程的国内外研究现状。第二章给出了全文的预备知识,对二次域、二次Euclid域中的重要理论和二次代数整数环Q ( D )中算术基本定理都有详细的介绍。第三章主要分六节具体证明了不定方程x~2 + D = 4y~5(D=7,11,-5,-13,-21,-29)的所有整数解。第一节证明了不定方程x~2 + 7 = 4y~5仅有整数解为( x , y )=(±11,2);第二节证明了不定方程x~2 + 11 = 4y~5仅有整数解为( x , y )=(±31,3);第三节证明了不定方程x~2 - 5 = 4y~5仅有整数解为( x , y )=(±1,-1)、(±3,1);第四节证明了不定方程x~2 - 13 = 4y~5仅有整数解为( x , y )=(±3,-1);第五节证明了不定方程x~2 - 21 = 4y~5仅有整数解为( x , y )=(±5,1);第六节证明了不定方程x~2 - 29 = 4y~5仅有整数解为( x , y )=(±5,-1)。第四章对全文作了一个总结,并对未来可能的发展方向提出了一些有待研究的问题。本文中的主要结果在第三章给出。更多还原

【Abstract】 The Diophantine equation not only developed actively itself, but also was apply to else fields of Discrete Mathematics. It plays an important role in people’s study and research to solve the actual problems. So many researchers study the Diophantine equation extensively and highly in the domestic and abroad. Along with the development of the Diophantine equation, Algebraic Number Theory obtained the first formation and developments. Currently, Algebraic Number Theory has become a branch of mathematics with abundant contents, is also an important tool of studying of the Diophantine equation.As an important part of Algebraic Number Theory, quadratic fields and the arithmetic in quadratic fields play an important role in research of the Diophantine equation. For some D , Q ( D ) is a Euclid area, And arithmetical theorem is carried out over the rings of quadratic integers. We can build up the same division theorem to solve some Diophantine equation. There are several work of study of solving the Diophantine equation x~2 + D = 4 pn with knowledge of Algebraic Number Theory, especially the study of the Diophantine equation x~2 + D = 4 pn according to discuss the divisibility of class number in imaginary quadratic fields. But it’s difficult to solve it in actual quadratic fields.In this paper, we will prove all the integer solutions of the Diophantine equation x~2 + D = 4y~5( D =7,11,-5,-13,-21,-29)with the method of some important theories in quadratic fields, the rings of quadratic integers Q ( D ) and Maple program in four parts. In the first part, we give the present condition of the Diophantine equation x~2 + D = 4 pn in the domestic and abroad. In the second part, we give the preparation knowledge of the whole paper. We introduced important theories in quadratic fields, quadratic Euclid area and the arithmetical theorem in the rings of quadratic integers Q ( D ) in detail. In the third part, we well prove that the Diophantine equation x~2 + 7 = 4y~5 has integer solutions ( x , y )=(±11,2);the Diophantine equation x~2 + 11 = 4y~5 has integer solutions ( x , y )=(±31,3);the Diophantine equation x~2 - 5 = 4y~5 has integer solutions ( x , y )=(±1,-1)and (±3,1);the Diophantine equation x~2 - 13 = 4y~5 has integer solutions ( x , y )=(±3, -1);the Diophantine equation x~2 - 21 = 4y~5 has integer solutions ( x , y )=(±5 ,1);the Diophantine equation x~2 - 29 = 4y~5 has integer solutions ( x , y )=(±5, -1).In the fourth part we summarize the whole paper, and put forward some problems which should be solved in perhaps development direction in the future.In this paper, main result will be given in the third part.更多还原