International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 7, Pages 1039-1047

Lagrange, central norms, and quadratic Diophantine equations

R. A. Mollin

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.


We consider the Diophantine equation of the form x2Dy2=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.