The Theory of Numbers

AUTHOR'S PREFACE.

THIS little book has been written from the point of view of the usefulness of the Theory of Relativity in the development of physical science. It is principally based on a short course of lectures which I delivered at Indiana University in the Fall of 1912 and on my recent papers in the Physical Review, vol. 35, pp. 153-176, and Second Series, vol. 1, pp. 161-197.

No attempt is made to give any applications of the theory other than what is incidental to the derivation of the fundamental results concerning length and time, the transformation of coordinates, mass and energy, and experimental verification. It is believed, however, that the presentation is such as to keep always close to concrete experience, so that the results obtained may be directly useful in suggesting experiments for the laboratory.

My indebtedness to other writers is indicated by the references in my recent papers referred to above. I wish also to speak of the useful suggestions made by my students in their frequent discussion of the Theory of Relativity during the course last Fall.

The purpose of this little book is to give the reader a convenient introduction to the theory of numbers, one of the most extensive and most elegant disciplines in the whole body of mathematics. The arrangement of the material is as follows: The first five chapters are devoted to the development of those elements which are essential to any study of the subject. The sixth and last chapter is intended to give the reader some indication of the direction of further study with a brief account of the nature of the material in each of the topics suggested. The treatment throughout is made as brief as is possible consistent with clearness and is confined entirely to fundamental matters. This is done because it is believed that in this way the book may best be made to serve its purpose as an introduction to the theory of numbers.

Numerous problems are supplied throughout the text. These have been selected with great care so as to serve as excellent exercises for the student's introductory training in the methods of number theory and to afford at the same time a further collection of useful results. The exercises marked with a star are more difficult than the others; they will doubtless appeal to the best students.

Finally, I should add that this book is made up from the material used by me in lectures in Indiana University during the past two years; and the selection of matter, especially of exercises, has been based on the experience gained in this way.