Characterization of a generalized Shanks sequence

We consider generalizations of Shanks’ sequence of quadratic fields ℚ( √Sn-- ) where S_n = (2ⁿ + 1)² + 2ⁿ⁺². Quadratic fields of this type are of interest because it is possible to explicitly determine the fundamental unit. If a sequence of quadratic fields given by D_n = A²x²ⁿ + Bxⁿ + C² satisfies certain conditions (notably that the regulator is of order Θ(n²)), then we determine the exact form such a sequence must take.

Keywords

periodic continued fraction, quadratic order, units

Mathematical Subject Classification 2000

Primary: 11R11, 11R27

Secondary: 11G20

Milestones

Received: 23 November 2004

Revised: 12 July 2006

Accepted: 7 December 2006

Published: 1 March 2007

Authors

Roger D. Patterson
Department of Mathematics Macquarie University Sydney, New South Wales Australia	Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4 Canada

Alfred J. van der Poorten
ceNTRe for Number Theory Research 1 Bimbil Pl. Killara, New South Wales 2071 Australia

Hugh C. Williams
Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4 Canada

Vol. 230, No. 1, 2007

Roger D. Patterson, Alfred J. van der Poorten and Hugh C. Williams

Abstract

Keywords

Mathematical Subject Classification 2000

Milestones

Authors