International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 7, Pages 1039-1047
doi:10.1155/IJMMS.2005.1039
Lagrange, central norms, and quadratic Diophantine equations
Department of Mathematics and Statistics, University of Calgary, Calgary T2N 1N4, AB, Canada
Received 4 May 2004; Revised 16 January 2005
Copyright © 2005 R. A. Mollin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider the Diophantine equation of the form x2−Dy2=c,
where c=±1,±2, 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, and elegant criterion for the central
norm to be 2 in the simple continued fraction expansion of
D.