Elementary Number Theory

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  • Edition: 7th
  • Format: Hardcover
  • Copyright: 2010-02-04
  • Publisher: McGraw-Hill Education

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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

Prefacep. viii
New to This Editionp. x
Preliminariesp. 01
Mathematical Inductionp. 01
The Binomial Theoremp. 08
Divisibility Theory in the Integersp. 13
Early Number Theoryp. 13
The Division Algorithmp. 17
The Greatest Common Divisorp. 19
The Euclidean Algorithmp. 26
The Diophantine Equation ax + by = cp. 32
Primes and Their Distributionp. 39
The Fundamental Theorem of Arithmeticp. 39
The Sieve of Eratosthenesp. 44
The Goldbach Conjecturep. 50
The Theory of Congruencesp. 61
Carl Friedrich Gaussp. 61
Basic Properties of Congruencep. 63
Binary and Decimal Representations of Integersp. 69
Linear Congruences and the Chinese Remainder Theoremp. 76
Fermat's Theoremp. 85
Pierre de Fermatp. 85
Fermat's Little Theorem and Pseudoprimesp. 87
Wilson's Theoremp. 93
The Fermat-Kraitchik Factorization Methodp. 97
Number-Theoretic Functionsp. 103
The Sum and Number of Divisorsp. 103
The Möbius Inversion Formulap. 112
The Greatest Integer Functionp. 117
An Application to the Calendarp. 122
Euler's Generalization of Fermat's Theoremp. 129
Leonhard Eulerp. 129
Euler's Phi-Functionp. 131
Euler's Theoremp. 136
Some Properties of the Phi-Functionp. 141
Primitive Roots and Indicesp. 147
The Order of an Integer Modulo np. 147
Primitive Roots for Primesp. 152
Composite Numbers Having Primitive Rootsp. 158
The Theory of Indicesp. 163
The Quadratic Reciprocity Lawp. 169
Euler's Criterionp. 169
The Legendre Symbol and Its Propertiesp. 175
Quadratic Reciprocityp. 185
Quadratic Congruences with Composite Modulip. 192
Introduction to Cryptographyp. 197
From Caesar Cipher to Public Key Cryptographyp. 197
The Knapsack Cryptosystemp. 209
An Application of Primitive Roots to Cryptographyp. 214
Numbers of Special Formp. 219
Marin Mersennep. 219
Perfect Numbersp. 221
Mersenne Primes and Amicable Numbersp. 227
Fermat Numbersp. 237
Certain Nonlinear Diophantine Equationsp. 245
The Equation x2 + y2 = z2p. 245
Fermat's Last Theoremp. 252
Representation of Integers as Sums of Squaresp. 261
Joseph Louis Lagrangep. 261
Sums of Two Squaresp. 263
Sums of More Than Two Squaresp. 272
Fibonacci Numbersp. 283
Fibonaccip. 283
The Fibonacci Sequencep. 285
Certain Identities Involving Fibonacci Numbersp. 292
Continued Fractionsp. 303
Srinivasa Ramanujanp. 303
Finite Continued Fractionsp. 306
Infinite Continued Fractionsp. 319
Farey Fractionsp. 334
Pell's Equationp. 337
Some Modern Developmentsp. 353
Hardy, Dickson, and Erdösp. 353
Primality Testing and Factorizationp. 358
An Application to Factoring: Remote Coin Flippingp. 371
The Prime Number Theorem and Zeta Functionp. 375
Miscellaneous Problemsp. 384
Appendixesp. 387
General Referencesp. 387
Suggested Further Readingp. 390
Tablesp. 393
Answers to Selected Problemsp. 410
Indexp. 421
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