Elementary Number Theory (Paperback) 7th Edition
Book Preface
The purpose of this volume is to give a simple account of classical number theory and to impart some of the historical background in which the subject evolved. Although primarily intended for use as a textbook in a one-semester course at the undergraduate level, it is designed to be used in teachers’ institutes or as supplementary reading in mathematics survey courses. The work is well suited for prospective secondary school teachers for whom a little familiarity with number theory may be particularly helpful.
The theory of numbers has always occupied a unique position in the world of mathematics. This is due to the unquestioned historical importance of the subject: it is one of the few disciplines having demonstrable results that predate the very idea of a university or an academy. Nearly every century since classical antiquity has witnessed new and fascinating discoveries relating to the properties of numbers; and, at some point in their careers, most of the great masters of the mathematical sciences have contributed to this body of knowledge. Why has number theory held such an irresistible appeal for the leading mathematicians and for thousands of amateurs? One answer lies in the basic nature of its problems. Although many questions in the field are extremely hard to decide, they can be formulated in terms simple enough to arouse the interest and curiosity of those with little mathematical training. Some of the simplest sounding questions have withstood intellectual assaults for ages and remain among the most elusive unsolved problems in the whole of mathematics. It therefore comes as something of a surprise to find that many students look upon number theory with good-humored indulgence, regarding it as a frippery on the edge of mathematics. This no doubt stems from the widely held view that it is the purest branch of pure mathematics and from the attendant suspicion that it can have few substantive applications to real-world problems. Some of the worst in order to digest the rest of the book. Problems whose solutions do not appear straightforward are frequently accompanied by hints.
The text was written with the mathematics major in mind; it is equally valuable for education or computer science majors minoring in mathematics. Very little is demanded in the way of specific prerequisites. A significant portion of the book can be profitably read by anyone who has taken the equivalent of a first-year college course in mathematics. Those who have had additional courses will generally be better prepared, if only because of their enhanced mathematical maturity. In particular, a knowledge of the concepts of abstract algebra is not assumed. When the book is used by students who have had an exposure to such matter, much of the first four chapters can be omitted.
Our treatment is structured for use in a wide range of number theory courses, of varying length and content. Even a cursory glance at the table of contents makes plain that there is more material than can be conveniently presented in an introductory one-semester course, perhaps even enough for a full-year course. This provides flexibility with regard to the audience and allows topics to be selected in accordance with personal taste. Experience has taught us that a semester-length course having the Quadratic Reciprocity Law as a goal can be built up from Chapters 1 through 9. It is unlikely that every section in these chapters need be covered; some or all of Sections 5.4, 6.2, 6.3, 6.4, 7.4, 8.3, 8.4, and 9.4 can be omitted from the program without destroying the continuity in our development. The text is also suited to serve a quarter-term course or a six-week summer session. For such shorter courses, segments of further chapters can be chosen after completing Chapter 4 to construct a rewarding account of number theory.
Chapters 10 through 16 are almost entirely independent of one another and so may be taken up or omitted as the instructor wishes. (Probably most users will want to continue with parts of Chapter 10, while Chapter 14 on Fibonacci numbers seems to be a frequent choice.) These latter chapters furnish the opportunity for additional reading in the subject, as well as being available for student presentations, seminars, or extra-credit projects.
Number theory is by nature a discipline that demands a high standard of rigor. Thus, our presentation necessarily has its formal aspect, with care taken to present clear and detailed arguments. An understanding of the statement of a theorem, not the proof, is the important issue. But a little perseverance with the demonstration will reap a generous harvest, for our hope is to cultivate the reader’s ability to follow a causal chain of facts, to strengthen intuition with logic. Regrettably, it is all too easy for some students to become discouraged by what may be their first intensive experience in reading and constructing proofs. An instructor might ease the way by approaching the beginnings of the book at a more leisurely pace, as well as restraining the urge to attempt all the interesting problems.
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