IJPAM: Volume 55, No. 1 (2009)

CENTRAL NORMS AND CONTINUED FRACTIONS

R.A. Moolin
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta, T2N 1N4, CANADA
e-mail: [email protected]
url: http://www.math.ucalgary.ca/~ ramollin/


Abstract.We look at solutions to the norm-form equations $x^2-Dy^2=c$ in terms of the central norm being equal to $c$ when the simple continued fraction expansion of $\sqrt{D}$ is odd. We essentially characterized all possible cases when the period length is even in earlier work including a generalization of a result of Lagrange for the case where $D$ is prime. However, virtually nothing has been done in the odd case. We make inroads herein that characterize the central norms in the odd case for a wide range of cases that link to what is known in the literature for the even case.

Received: January 11, 2008

AMS Subject Classification: 11A55, 11D09, 11R11

Key Words and Phrases: quadratic Diophantine equations, continued fractions, central norms, fundamental units

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 1