Date: Fri, 17 Nov 1995 13:45:40 +0500
From: nsinger@eos.hitc.com (Nick Singer)
Subject: Your SOLVING INTEGRAL QUADRATICS INTEGRALLY AND RATIONALLY
To: rusin@math.niu.edu

I greatly enjoyed downloading and reading your article SOLVING INTEGRAL 
QUADRATICS INTEGRALLY AND RATIONALLY.

You say "I am not familiar with the methods used for the computation of the 
class group of real quadratic fields, nor the computation of which element 
of the group each ideal pertains to.  Thus I can't give an effective 
solution to the general Pellian equation.  Surely, however, this is known.  
One can show that if any solution exists (with x1>0, y1>0, say) then one 
exists with x1+y1Sqrt(D2) < Sqrt(N0.e), where  e  is the fundamental unit.  
See [Borevich & Shafarevich, p. 123].  I believe there is also some sort of 
reduction which reduces the question for general  N0  to those N0 < 
2sqrt(D), which is then addressed with continued fractions, but I can't 
remember what it is."

An excellent elementary reference for this is G. Chrystal, Textbook of 
Algebra, vol. 2, chapter XXXIII, sections 15-20.  This is (re)published by 
Dover in paperback; it was originally published in 1900.  It has a very good 
exposition of continued fractions (chaps XXXII thru XXXIV), and addresses 
(in chapter XXXIII, section 20) the very subject of your paper.

Cheers,
Nick

PS. The first equation in your Sect. V wants to have a plus sign in place of 
the minus sign.