Burton D. Elementary Number Theory

7th ed. — McGraw-Hill Science/Engineering/Math, 2010. — 448 p. — ISBN: 0073383147, 9780073383149.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.New to this Edition
Preliminaries
Mathematical Induction
The Binomial Theorem
Divisibility Theory in the Integers
Early Number Theory
The Division Algorithm
The Greatest Common Divisor
The Euclidean Algorithm
The Diophantine Equation
Primes and Their Distribution
The Fundamental Theorem of Arithmetic
The Sieve of Eratosthenes
The Goldbach Conjecture
The Theory of Congruences
Carl Friedrich Gauss
Basic Properties of Congruence
Binary and Decimal Representations of Integers
Linear Congruences and the Chinese Remainder Theorem
Fermat’s Theorem
Pierre de Fermat
Fermat’s Little Theorem and Pseudoprimes
Wilson’s Theorem
The Fermat-Kraitchik Factorization Method
Number-Theoretic Functions
The Sum and Number of Divisors
The Möbius Inversion Formula
The Greatest Integer Function
An Application to the Calendar
Euler’s Generalization of Fermat’s Theorem
Leonhard Euler
Euler’s Phi-Function
Euler’s Theorem
Some Properties of the Phi-Function
Primitive Roots and Indices
The Order of an Integer Modulo n
Primitive Roots for Primes
Composite Numbers Having Primitive Roots
The Theory of Indices
The Quadratic Reciprocity Law
Euler’s Criterion
The Legendre Symbol and Its Properties
Quadratic Reciprocity
Quadratic Congruences with Composite Moduli
Introduction to Cryptography
From Caesar Cipher to Public Key Cryptography
The Knapsack Cryptosystem
An Application of Primitive Roots to Cryptography
Numbers of Special Form
Marin Mersenne
Perfect Numbers
Mersenne Primes and Amicable Numbers
Fermat Numbers
Certain Nonlinear Diophantine Equations
The Equation
Fermat’s Last Theorem
Representation of Integers as Sums of Squares
Joseph Louis Lagrange
Sums of Two Squares
Sums of More Than Two Squares
Fibonacci Numbers
Fibonacci
The Fibonacci Sequence
Certain Identities Involving Fibonacci Numbers
Continued Fractions
Srinivasa Ramanujan
Finite Continued Fractions
Infinite Continued Fractions
Farey Fractions
Pell’s Equation
Some Recent Developments
Hardy, Dickson, and Erdös
Primality Testing and Factorization
An Application to Factoring: Remote Coin Flipping
The Prime Number Theorem and Zeta Function
Miscellaneous Problems
Appendixes
General References
Suggested Further Reading
Tables
Answers to Selected Problems