Elementary Number Theory in Nine Chapters

by James J. Tattersall

• ISBN13: 9780521615242
• ISBN10: 0521615240
• eBook ISBN(s): 9780511123870
• Edition: 2nd
• Format: Paperback
• Copyright: 2005-07-25
• Publisher: Cambridge University Press
• More Book Details

Summary

This textbook is intended to serve as a one-semester introductory course in number theory and in this second edition it has been revised throughout and many new exercises have been added. Historical perspective is included and emphasis is given to some of the subject's applied aspects; in particular the field of cryptography is highlighted. At the heart of the book are the major number theoretic accomplishments of Euclid, Fermat, Gauss, Legendre, and Euler, and to fully illustrate the properties of numbers and concepts developed in the text, a wealth of exercises have been included. It is assumed that the reader will have 'pencil in hand' and ready access to a calculator or computer. For students new to number theory, whatever their background, this is a stimulating and entertaining introduction to the subject.

Table of Contents

Preface	ix
The intriguing natural numbers
Polygonal numbers	1	(22)
Sequences of natural numbers	23	(17)
The principle of mathematical induction	40	(3)
Miscellaneous exercises	43	(7)
Supplementary exercises	50	(5)
Divisibility
The division algorithm	55	(9)
The greatest common divisor	64	(6)
The Euclidean algorithm	70	(6)
Pythagorean triples	76	(5)
Miscellaneous exercises	81	(3)
Supplementary exercises	84	(3)
Prime numbers
Euclid on primes	87	(7)
Number theoretic functions	94	(9)
Multiplicative functions	103	(5)
Factoring	108	(4)
The greatest integer function	112	(3)
Primes revisited	115	(14)
Miscellaneous exercises	129	(4)
Supplementary exercises	133	(3)
Perfect and amicable numbers
Perfect numbers	136	(9)
Fermat numbers	145	(2)
Amicable numbers	147	(3)
Perfect-type numbers	150	(9)
Supplementary exercises	159	(2)
Modular arithmetic
Congruence	161	(8)
Divisibility criteria	169	(4)
Euler's phi-function	173	(8)
Conditional linear congruences	181	(9)
Miscellaneous exercises	190	(3)
Supplementary exercises	193	(3)
Congruences of higher degree
Polynomial congruences	196	(4)
Quadratic congruences	200	(12)
Primitive roots	212	(10)
Miscellaneous exercises	222	(1)
Supplementary exercises	223	(3)
Cryptology
Monoalphabetic ciphers	226	(9)
Polyalphabetic ciphers	235	(10)
Knapsack and block ciphers	245	(5)
Exponential ciphers	250	(5)
Supplementary exercises	255	(3)
Representations
Sums of squares	258	(16)
Pell's equation	274	(6)
Binary quadratic forms	280	(3)
Finite continued fractions	283	(8)
Infinite continued fractions	291	(7)
p-Adic analysis	298	(4)
Supplementary exercises	302	(2)
Partitions
Generating functions	304	(2)
Partitions	306	(5)
Pentagonal Number Theorem	311	(13)
Supplementary exercises	324	(2)
Tables
List of symbols used	326	(3)
Primes less than 10 000	329	(4)
The values of τ(n), σ(n), φ(n), μ(n), ω(n), and Ω(n) for natural numbers less than or equal to 100	333	(3)
Answers to selected exercises	336	(85)
Bibliography
Mathematics (general)	411	(1)
History (general)	412	(1)
Chapter references	413	(8)
Index	421