Constructing families of long continued fractions

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.

Some more long continued fractions, I

“…This example generalizes Madden's second example (p. 130 of [11]) (set m = 1) (Madden's second example can also be found in row 2 of Table 3 The example given by van der Poorten in [17], namely…”

“…This long continued fraction generalizes Madden's first example in Section 3 of [11], where Madden's continued fraction is the case m = 1 of the continued fraction above (this continued fraction of Madden is also given in row 1 of Table 3 in Williams' paper [23]…”

“…Some general propositions. We next state several variants of a result of Madden from [11]. These general propositions will allow us to construct specific families of long continued fractions in the next section.…”

“…These include those discovered by Hendy [6], Bernstein [2] and [3], Williams [23], [22] and [21] , Levesque and Rhin [8], Azuhatu [1], Levesque [7], Halter-Koch [4] and [5], Mollin and Williams [13] and [14], van der Poorten [17], and, more recently, Madden [11] and Mollin [12].…”

“…In this present paper we use matrix methods (based on Madden's method in [11]) to derive several infinite families of quadratic surds with long continued fraction expansions. We feel that, while these matrix methods may not be as widely applicable, where they can be used the proofs are more transparent, more direct and less intricate than proofs by the method outlined above in the papers of Mollin and Williams.…”