New York Journal of Mathematics
Volume 11 (2005) 539-545


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

We consider the Diophantine equation of the form x2-Dy2=\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}.


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