Constructing families of long continued fractions

Daniel J. Madden

Vol. 198 (2001), No. 1, 123–147

DOI: 10.2140/pjm.2001.198.123

Abstract

This paper describes a method of constructing an unlimited number of infinite families of continued fraction expansions of the square root of D, an integer. The periods of these continued fractions all have identifiable sub patterns repeated a number of times according to certain parameters. For example, it is possible to construct an explicit family for the square root of D(k,l) where the period of the continued fraction has length 2kl − 2. The method is recursive and additional parameters controlling the length can be added.

Milestones

Received: 9 October 1998

Revised: 15 November 1999

Published: 1 March 2001