关于几类不定方程的整数解

发布时间：2018-01-23 08:14

本文关键词： 不定方程整数解代数数论二次域同余　出处：《重庆师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：不定方程是数论中的一个非常重要的领域,它有着悠久的历史和丰富的内容.所谓不定方程是指未知数为正整数、整数、有理数或代数整数等的方程或方程组,其未知数的个数一般多于方程式的个数.它在代数数论、组合设计、有限单群等领域都有着广泛的应用.本文主要工作在于利用代数数论的有关知识,讨论了以下不定方程：一、讨论了不定方程x~2+D=y~3.(D=-14,-42)的整数解,证明了1.不定方程x~2-14=y~3无整数解；2.不定方程x~2-42=y~3无整数解.二、讨论了不定方程x~2+D=4y~3(D=19,43,67,15,35,51,79,71,-69)的整数解,证明了1.不定方程x~2+D=4y~3,D=19,43,67无整数解；2.不定方程x~2+D=4y~3,D=15,35,51无整数解；3.不定方程x~2+79=4y~3仅有整数解(±265,26)；4.不定方程x~2+71=4y~3仅有整数解(±235,24)；5.不定方程x~2-69=4y~3无整数解.三、讨论不定方程x~2+D=4y~5(D=19,35,51,91,31,59,83)的整数解,证明了1.不定方程x~2+19=4y~5无整数解；2.不定方程x~2+D=4y~5,D=35,51,91无整数解；3.不定方程x~2+D=4y~5,D=31,59,83无整数解.
[Abstract]:Diophantine equation is a very important field in number theory. It has a long history and rich contents. The so-called indefinite equation refers to the equation or system of equations in which unknown numbers are positive integers, integers, rational numbers or algebraic integers. The number of unknown numbers is generally more than the number of equations. It has a wide range of applications in the fields of algebraic number theory, combinatorial design, finite simple group, etc. The main work of this paper is to use the knowledge of algebraic number theory. The following indeterminate equations are discussed: 1. The integer solution of the indeterminate equation x ~ (2) D ~ (~ (2) D) ~ ((14) ~ (-42)) is discussed, and it is proved that (1) there is no integer solution to the indeterminate equation x _ (2) ~ (14) y ~ (3); 2. There is no integer solution to the indeterminate equation x ~ (2) ~ (2) ~ (2) ~ (2) ~ (2). Secondly, it is discussed that the indeterminate equation x ~ (2) D ~ (2) D ~ ((4)) ~ (4) ~ (3) ~ (3) D ~ (19) ~ (1) ~ (3) ~ (3) ~ (. It is proved that there is no integer solution for the indeterminate equation x ~ (2) D ~ (2) D ~ (4) ~ (3) ~ ~ (1) ~ (3) ~ (6 ~ (6)); 2. There is no integer solution to the indeterminate equation x ~ (2) D ~ (2) D ~ (4) y ~ (3) ~ (3) ~ (1) ~ (1) ~ (3). 3. The indeterminate equation x ~ (2 79) ~ (4) y ~ (3) has only an integer solution (卤265 ~ (26)); 4. The indeterminate equation x _ (271n) ~ (4) y ~ (3) has only an integer solution (卤235n ~ (24)); 5. There is no integer solution for the indeterminate equation x ~ (2-69) ~ 4yn _ 3. Thirdly, the integer solution of the indeterminate equation x ~ (2) D ~ (2) D ~ (4) ~ (4) ~ (5) D ~ (1) ~ (1) ~ (1) ~ (1) ~ (1) ~ (1)) is discussed. It is proved that 1. There is no integer solution to the indeterminate equation x ~ (21) ~ (19) ~ (4) y ~ (5); 2. There is no integer solution to the indeterminate equation x ~ (2) D ~ (2) D ~ (4) ~ (4) y ~ (?) ~ (?). 3. There is no integer solution for the indeterminate equation x ~ (2) D ~ (2) D ~ (4) y ~ (?) ~ (?) ~ (?)
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O156

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