Synopses & Reviews
An undergraduate-level introduction to number theory, with the emphasis on fully explained proofs and examples. Exercises, together with their solutions are integrated into the text, and the first few chapters assume only basic school algebra. Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares. In particular, the last chapter gives a concise account of Fermat's Last Theorem, from its origin in the ancient Babylonian and Greek study of Pythagorean triples to its recent proof by Andrew Wiles.
Review
From the reviews:
BULLETIN OF MATHEMATICS BOOKS
?as a nice concluding chapter on Fermat? Last Theorem, with a brief discussion on the coup de grace."
Review
From the reviews: BULLETIN OF MATHEMATICS BOOKS "?as a nice concluding chapter on Fermat? Last Theorem, with a brief discussion on the coup de grace." G.A. Jones and J.M. Jones Elementary Number Theory "A welcome addition . . . a carefully and well-written book."--THE MATHEMATICAL GAZETTE "This book would make an excellent text for an undergraduate course on number theory." --MATHEMATICAL REVIEWS
Synopsis
Our intention in writing this book is to give an elementary introduction to number theory which does not demand a great deal of mathematical back ground or maturity from the reader, and which can be read and understood with no extra assistance. Our first three chapters are based almost entirely on A-level mathematics, while the next five require little else beyond some el ementary group theory. It is only in the last three chapters, where we treat more advanced topics, including recent developments, that we require greater mathematical background; here we use some basic ideas which students would expect to meet in the first year or so of a typical undergraduate course in math ematics. Throughout the book, we have attempted to explain our arguments as fully and as clearly as possible, with plenty of worked examples and with outline solutions for all the exercises. There are several good reasons for choosing number theory as a subject. It has a long and interesting history, ranging from the earliest recorded times to the present day (see Chapter 11, for instance, on Fermat's Last Theorem), and its problems have attracted many of the greatest mathematicians; consequently the study of number theory is an excellent introduction to the development and achievements of mathematics (and, indeed, some of its failures). In particular, the explicit nature of many of its problems, concerning basic properties of inte gers, makes number theory a particularly suitable subject in which to present modern mathematics in elementary terms."
Synopsis
This is an elementary undergraduate level introduction to number theory, with carefully explained proofs and numerous exercises and worked examples. A feature of particular interest is a concise account of Fermat's 'Last Theorem' and its recent proof by Andrew Wiles.
Table of Contents
Preface.- Notes to the reader.- Divisibility.- Prime Numbers.- Congruences.- Congruences with prime modulus.- Euler's function.- The group of units.- Quadratic residues.- Arithmetic functions.- The Riemann zeta function.- Sums of squares.- Fermat's Last Theorem.- Appendix 1: Induction and well-ordering.- Appendix 2: Groups, rings and fields.- Appendix 3: Convergence.- Table of primes.- Solutions to excercises.- References.- Index of symbols.- Index of names.- Index.