From: steiner@math.bgsu.edu (ray steiner)
Subject: Re: "Number Theorie: only one solution?" Some comments and questions
Date: Tue, 24 Aug 1999 14:42:53 -0500
Newsgroups: sci.math
Keywords: How well do continued fractions behave for  sqrt(D) ?

More questions:
     Let D be a positive, square free integer.
1). Suppose x, y is the smallest positive integer solution of x^2-Dy^2=1.
Then we know, x/y must be a convergent in the continued fraction expansion 
of sqrt(D).
and, in fact, that  |x/y -sqrt(D)| < 1/(2y^2) .
Questions: 
1).We know all other positive integer solutions are given by
a_n +b_n*sqrt(D)= (x + y*sqrt(D))^n.
How closely do these other solutions approximate sqrt(D)?

2). Are the partial quotients in the continued fraction expansion of sqrt(D)
bounded by some multiple of sqrt(D)?
These seem to be important questions, but none of my textbooks address them!
Regards,
Ray Steiner

-- 
steiner@math.bgsu.edu
==============================================================================

From: dredmond@math.siu.edu (Don Redmond)
Subject: Re: "Number Theorie: only one solution?" Some comments and questions
Date: Tue, 24 Aug 1999 18:21:20 -0500
Newsgroups: sci.math

In article <steiner-2408991442530001@math-38.bgsu.edu>,
steiner@math.bgsu.edu (ray steiner) wrote:

> More questions:
>      Let D be a positive, square free integer.
> 1). Suppose x, y is the smallest positive integer solution of x^2-Dy^2=1.
> Then we know, x/y must be a convergent in the continued fraction expansion 
> of sqrt(D).
> and, in fact, that  |x/y -sqrt(D)| < 1/(2y^2) .
> Questions: 
> 1).We know all other positive integer solutions are given by
> a_n +b_n*sqrt(D)= (x + y*sqrt(D))^n.
> How closely do these other solutions approximate sqrt(D)?
> 
> 2). Are the partial quotients in the continued fraction expansion of sqrt(D)
> bounded by some multiple of sqrt(D)?
> These seem to be important questions, but none of my textbooks address them!
> Regards,
> Ray Steiner

As for (1) I think the answer is that they get better. Aren't they all found
in the convergents for sqr(D)? I can't remember.

As for (2) I pretty sure the answe ris yes and is, for example, an exercise
in the appropiate chapter of Chrystal's Algebra.

Don