关于几类不定方程组整数解的研究

【作者】 高洁；

【导师】 袁进；

【作者基本信息】 西北大学 ， 基础数学， 2011， 硕士

【摘要】 不定方程是数论中的一个重要分支,其内容极为广泛,与代数数论和代数几何等数学其他分支联系紧密.关于不定方程的一些结论在丢番图逼近论,代数数论,P-adic等相关学科中有着广泛应用.不定方程及不定方程组的研究大大充实了数论研究的内容.本文研究的几类不定方程组,已经有很多学者进行研究,但由于解不定方程的困难性,还有很大的研究空间.本文运用数论研究中的一些初等方法讨论了几类不定方程组及不定方程的求解问题,主要内容：1.不定方程组(其中p为奇素数)无正整数解的一些充分条件.2.不定方程组(其中p为奇素数)无正整数解的一些充分条件.3.不定方程X3±1=py2(其中p为奇素数)无正整数解的一些充分条件,证明了存在无数个6k+1型的素数p使得方程X3±1=py2无正整数解.4.不定方程组(D是无平方因子的正整数)无正整数解的充分条件.更多还原

【Abstract】 As an important branch of number theory, the indeterminate equation has comprehensive content and correlates closely to algebraic number the-ory, algebraic geometry and other parts of math. Some conclusion about the indeterminate equation are applied to Diophantine approximation, algebraic ge-ometry, p-adic and other relevant subjects. the research on the indeterminate equation have enriched the content of number theory widely.Although many scholars have research on the indeterminate equations, there is still a large research space on these problems.so that this thesis uses elemen-tary methods in theory of numbers to discuss and analyze several kinds of the indeterminate equation (equations) and reaches conclusions specifically:1. Some sufficient conditions are obtained that the indeterminate equations has no positive integer solution,where p is odd prime.2. Some sufficient conditions are obtained that the indeterminate equations has no positive integer solution,where p is odd prime.3. Some sufficient conditions are obtained that the indeterminate equation x3±1=py2 has no positive integer solution,where p is odd prime. And get countless p, where p is an odd prime of the form 6k+1, meet the equation x3±1=py2 without positive integer solution.4. A sufficient conditions is obtained that the indeterminate equations has no positive integer solution,where D is a positive integer with square free.更多还原