R.A. Mollin, Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada, T2N 1N4; email: ramollin@math.ucalgary.ca

Abstract/Résumé:

Central norms are given definition according to the infrastructure of the underlying order under discussion, which we define in the introductory section below. We relate these central norms in the simple continued fraction expansion of \(\sqrt{D}\) to solutions of the Eisenstein equation \(x^2-Dy^2 = -4\), with \(\gcd(x,y) = 1\). This provides a criterion for central norms to be \(4\) in the presence of certain congruence conditions on the fundamental unit of the underlying real quadratic order \(\mathbb{Z}[\sqrt{D}]\).

Keywords: Eisenstein equations, central norms, continued fractions

AMS Subject Classification: Quadratic and bilinear equations 11D09

PDF(click to download): Eisenstein Equations and Central Norms