用佩尔方程x^2-dy^2=1的分数解求其整数解

Using Fraction Solutions of Pell FangCheng x^2-dy^2=1 to Extract Its Integer Ones

求佩尔方程x^2-dy^2=1的全部整数解,关键是如何求出其基本解。求基本解的方法有两种:一是用试验的方法寻求基本解;二是求基本解的一般方法,把d^(1/2)展成简单连分数后进行求解。用这两种方法求解一般过程较繁。佩尔方程不但有无穷多组整数解,也有无穷多组分数解。求佩尔方程(1)的分数解易于求其整数解。如能用分数解求出整数解,这样会使问题变的简单。本文就是基于这种想法,总结出一种用分数解求基本解的方法。

It is critical that we extract elementary solutions of Pell equation when we want to get its integer ones. Generally there are two ways to extract elementary solutions; testing and continued fractioning. The two processes are complex. Not only dose Pell equation have infinite sets of integer solutions,but it also has infinite fraction ones,the latter is easier than the former. If we can use the fraction solutions of Pell equation to get its integer ones, it will be simpler for us to do the extraction. This article is to summarize the method of using frac- tion solutions to extract its integer ones.