Elementary Number Theory

by Burton, David

• ISBN13: 9780073383149
• ISBN10: 0073383147
• Edition: 7th
• Format: Hardcover
• Copyright: 2010-02-04
• Publisher: McGraw Hill

Summary

Elementary Number Theory, Seventh Edition, is written for the one-semester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject's evolution from antiquity to recent research. Written in David Burton's engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history.

Table of Contents

Preface	p. viii
New to This Edition	p. x
Preliminaries	p. 01
Mathematical Induction	p. 01
The Binomial Theorem	p. 08
Divisibility Theory in the Integers	p. 13
Early Number Theory	p. 13
The Division Algorithm	p. 17
The Greatest Common Divisor	p. 19
The Euclidean Algorithm	p. 26
The Diophantine Equation ax + by = c	p. 32
Primes and Their Distribution	p. 39
The Fundamental Theorem of Arithmetic	p. 39
The Sieve of Eratosthenes	p. 44
The Goldbach Conjecture	p. 50
The Theory of Congruences	p. 61
Carl Friedrich Gauss	p. 61
Basic Properties of Congruence	p. 63
Binary and Decimal Representations of Integers	p. 69
Linear Congruences and the Chinese Remainder Theorem	p. 76
Fermat's Theorem	p. 85
Pierre de Fermat	p. 85
Fermat's Little Theorem and Pseudoprimes	p. 87
Wilson's Theorem	p. 93
The Fermat-Kraitchik Factorization Method	p. 97
Number-Theoretic Functions	p. 103
The Sum and Number of Divisors	p. 103
The Möbius Inversion Formula	p. 112
The Greatest Integer Function	p. 117
An Application to the Calendar	p. 122
Euler's Generalization of Fermat's Theorem	p. 129
Leonhard Euler	p. 129
Euler's Phi-Function	p. 131
Euler's Theorem	p. 136
Some Properties of the Phi-Function	p. 141
Primitive Roots and Indices	p. 147
The Order of an Integer Modulo n	p. 147
Primitive Roots for Primes	p. 152
Composite Numbers Having Primitive Roots	p. 158
The Theory of Indices	p. 163
The Quadratic Reciprocity Law	p. 169
Euler's Criterion	p. 169
The Legendre Symbol and Its Properties	p. 175
Quadratic Reciprocity	p. 185
Quadratic Congruences with Composite Moduli	p. 192
Introduction to Cryptography	p. 197
From Caesar Cipher to Public Key Cryptography	p. 197
The Knapsack Cryptosystem	p. 209
An Application of Primitive Roots to Cryptography	p. 214
Numbers of Special Form	p. 219
Marin Mersenne	p. 219
Perfect Numbers	p. 221
Mersenne Primes and Amicable Numbers	p. 227
Fermat Numbers	p. 237
Certain Nonlinear Diophantine Equations	p. 245
The Equation x2 + y2 = z2	p. 245
Fermat's Last Theorem	p. 252
Representation of Integers as Sums of Squares	p. 261
Joseph Louis Lagrange	p. 261
Sums of Two Squares	p. 263
Sums of More Than Two Squares	p. 272
Fibonacci Numbers	p. 283
Fibonacci	p. 283
The Fibonacci Sequence	p. 285
Certain Identities Involving Fibonacci Numbers	p. 292
Continued Fractions	p. 303
Srinivasa Ramanujan	p. 303
Finite Continued Fractions	p. 306
Infinite Continued Fractions	p. 319
Farey Fractions	p. 334
Pell's Equation	p. 337
Some Modern Developments	p. 353
Hardy, Dickson, and Erdös	p. 353
Primality Testing and Factorization	p. 358
An Application to Factoring: Remote Coin Flipping	p. 371
The Prime Number Theorem and Zeta Function	p. 375
Miscellaneous Problems	p. 384
Appendixes	p. 387
General References	p. 387
Suggested Further Reading	p. 390
Tables	p. 393
Answers to Selected Problems	p. 410
Index	p. 421
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