Elementary Number Theory and Its Applications

by
Edition: 6th
Format: Hardcover
Pub. Date: 2011-01-01
Publisher(s): Addison Wesley
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Summary

Elementary Number Theory and Its Applicationsis noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the mathematical content and the exercises. The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.

Table of Contents

What Is Number Theory? 1(4)
The Integers
5(38)
Numbers and Sequences
6(10)
Sums and Products
16(7)
Mathematical Induction
23(7)
The Fibonacci Numbers
30(7)
Divisibility
37(6)
Integer Representations and Operations
43(24)
Representations of Integers
43(10)
Computer Operations with Integers
53(7)
Complexity of Integer Operations
60(7)
Primes and Greatest Common Divisors
67(74)
Prime Numbers
68(9)
The Distribution of Primes
77(13)
Greatest Common Divisors
90(7)
The Euclidean Algorithm
97(11)
The Fundamental Theorem of Arithmetic
108(15)
Factorization Methods and the Fermat Numbers
123(10)
Linear Diophantine Equations
133(8)
Congruences
141(48)
Introduction to Congruences
141(12)
Linear Congruences
153(5)
The Chinese Remainder Theorem
158(10)
Solving Polynomial Congruences
168(6)
Systems of Linear Congruences
174(10)
Factoring Using the Pollard Rho Method
184(5)
Applications of Congruences
189(26)
Divisibility Tests
189(6)
The Perpetual Calendar
195(5)
Round-Robin Tournaments
200(2)
Hashing Functions
202(5)
Check Digits
207(8)
Some Special Congruences
215(24)
Wilson's Theorem and Fermat's Little Theorem
215(8)
Pseudoprimes
223(10)
Euler's Theorem
233(6)
Multiplicative Functions
239(38)
The Euler Phi-Function
239(11)
The Sum and Number of Divisors
250(7)
Perfect Numbers and Mersenne Primes
257(12)
Mobius Inversion
269(8)
Cryptology
277(56)
Character Ciphers
278(8)
Block and Stream Ciphers
286(19)
Exponentiation Ciphers
305(3)
Public Key Cryptography
308(8)
Knapsack Ciphers
316(7)
Cryptographic Protocols and Applications
323(10)
Primitive Roots
333(46)
The Order of an Integer and Primitive Roots
334(7)
Primitive Roots for Primes
341(6)
The Existence of Primitive Roots
347(8)
Index Arithmetic
355(10)
Primality Tests Using Orders of Integers and Primitive Roots
365(7)
Universal Exponents
372(7)
Applications of Primitive Roots and the Order of an Integer
379(22)
Pseudorandom Numbers
379(10)
The ElGamal Cryptosystem
389(5)
An Application to the Splicing of Telephone Cables
394(7)
Quadratic Residues
401(54)
Quadratic Residues and Nonresidues
402(15)
The Law of Quadratic Reciprocity
417(13)
The Jacobi Symbol
430(9)
Euler Pseudoprimes
439(9)
Zero-Knowledge Proofs
448(7)
Decimal Fractions and Continued Fractions
455(54)
Decimal Fractions
455(13)
Finite Continued Fractions
468(10)
Infinite Continued Fractions
478(12)
Periodic Continued Fractions
490(14)
Factoring Using Continued Fractions
504(5)
Some Nonlinear Diophantine Equations
509(38)
Pythagorean Triples
510(6)
Fermat's Last Theorem
516(12)
Sums of Squares
528(11)
Pell's Equation
539(8)
The Gaussian Integers
547(30)
Gaussian Integers and Gaussian Primes
547(12)
Greatest Common Divisors and Unique Factorization
559(11)
Gaussian Integers and Sums of Squares
570(7)
A Axioms for the Set of Integers
577(4)
B Binomial Coefficients
581(8)
C Using Maple and Mathematica for Number Theory
589(10)
Using Maple for Number Theory
589(4)
Using Mathematica for Number Theory
593(6)
D Number Theory Web Links
599(2)
E Tables
601(16)
Answers to Odd-Numbered Exercises 617(72)
Bibliography 689(14)
Index of Biographies 703(2)
Index 705(16)
Photo Credits 721

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