摘要
本文用模型论和数论方法讨论整数环的某些扩环上一些丢番图方程素元解的问题.这里讨论的丢番图方程有三类:1、Pell方程x^2-dy^2=1,d是不等于零的有理整数.2、Mordell方程x^3+y^2=6.3、Fermat万程x^n+y^n=Z^n.证明了它们中某些具有素元解,有些不具有素元解,本文说明用模型论来讨论整数环的扩环具有一定的意义.本文所用的模型论方面的知识主要是超积的概念及一些基本性质,可参考文献〔1〕.
The intention of the paper is to discuss the prime number solutions to some Diophantine equations on certain extention rings of ring of integers by model-theoretic and number-theoretic mothods. In this paper we deal with three kinds of equations : ( 1 ) the Pell equation x2-dy2=1, d≠0; (2) the Mordell equation x3 + y2 = 6; (3) the Fermat equation x+y =z . We prove that some of them have prime number solutions (infinitely many) and some of them do not.
关键词
整数环
丢番图方程
模型论
超积
model theory
ultraproduct
Diophantine equation
prime solution
