 

R.A. Mollin
Generalized Lagrange criteria for certain quadratic Diophantine equations


Published: 
November 8, 2005

Keywords: 
Quadratic Diophantine equations, Continued Fractions, Central Norms 
Subject: 
Primary: 11D09, 11R11, 11A55. Secondary: 11R29 


Abstract
We consider the Diophantine equation of the form
x^{2}Dy^{2}=\pm4, where D is a positive integer that is
not a perfect square, and provide a
generalization of results of Lagrange with elementary proofs
using only basic properties of simple continued fractions. As a
consequence, we achieve a completely general,
simple
criterion for the central norm to be 4
associated with
principal norm 8 in the simple continued
fraction expansion of
\sqrt{D}.


Acknowledgements
The author's research is supported by NSERC Canada grant # A8484.


Author information
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4, Canada
ramollin@math.ucalgary.ca
http://www.math.ucalgary.ca/~ramollin/

