Number Theory

and Its History

Oystein Ore


This book is based upon a course dealing with the theory of numbers and its history which has been given at Yale for several years. Although the course has been attended primarily by college students in their junior and senior years it has been open to all interested. The lectures were intended to give the principal ideas and methods of number theory as well as their historical background and development through the centuries. Most texts on number theory contain inserted historical notes but in this course I have attempted to obtain a presentation of the results of the theory integrated more fully in the historical and cultural framework. Number theory seems particularly suited to this form of exposition, and in my experience it has contributed much to making the subject more informative as well as more palatable to the students.

Obviously, only some of the main problems of number theory could be included in this book. In making a selection, topics of systematic and historical importance capable of a simple presentation have been preferred. While many standard aspects of number theory had to be discussed, the treatment is often new, and much material has been added that has not heretofore made its appearance in texts. Also, in several instances I have found it desirable to introduce and define modern algebraic concepts whose usefulness is readily explained by the context.

The questions of number theory are of importance not only to mathematicians. Now, as in earlier days, these problems seem to possess a particular attraction for many laymen, and number theory is notable as one of the few fields of mathematics where the suggestions and conjectures of amateurs or nonprofessional mathematicians have exerted an appreciable influence. It may be mentioned incidentally that there have been few college classes that I can recall in which there were not to be found some students who had already played with the strange properties of numbers. To make. the theory available to readers whose mathematical knowledge may be limited, every effort has been made to reduce to a minimum the technical complications and mathematical requirements of the presentation. Thus, the book is of a more elementary character than many previous texts, and for the understanding of a greater part of the subject matter a knowledge of the simplest algebraic rules should be sufficient. Only in some of the later chapters has a more extended familiarity with mathematical manipulations been presupposed.

August, 1948


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