Arithmetical continued fractions
Bankier, James Douglas
Doctor of Philosophy
The study of arithmetical continued fractions has been restricted, for the most part, to the investigation of the properties of regular continued fractions and questions of convergence. An important exception to this statement is a paper of Leighton's in which he introduced the concept of proper continued fractions, and the greater part of this thesis is devoted to further investigations in this field. Proper continued fractions are not only of interest themselves, but they throw considerable light upon regular continued fractions as well. The partial numerators of regular continued fractions are rather anonymous in character, since they are all equal to unity, and, in the classical presentation of regular continued fractions, it is not always clear just what part the partial numerators play in the investigation. This is not the case with proper continued fractions, and, as a result of the deeper insight which this fact made possible, it was found that many of the classical theorems about regular continued fractions were concerned with properties which were not essentially characteristic of regular continued fractions. A striking example of this point of view is the generalization of Galois' theorem which is presented in this thesis. The generalized theorem depends only upon the periodic character of the continued fractions with which the theorem is concerned, and the fact that the continued fractions are convergent. This thesis has been divided into four chapters. The first chapter deals with conditions that a number be equal to an approximant of a proper continued fraction, and considers to what extent these approximants may be regarded as best approximations. In the second chapter a more extensive class of continued fractions, whose elements are complex numbers, is considered, and the results of this investigation are used in the study of periodic proper continued fractions. A new method of determining the elements of a continued fraction, whose approximants are a preassigned sequence of numbers, is given in Chapter III, and the method is used to determine conditions for the periodicity of a proper continued fraction expansion. In the fourth chapter certain known facts concerning the Pell equation are obtained by a method which is new.