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.



Preface 

New to this Edition 
1 Preliminaries 
1.1 Mathematical Induction 
1.2 The Binomial Theorem 

2 Divisibility Theory in the Integers 
2.1 Early Number Theory 
2.2 The Division Algorithm 
2.3 The Greatest Common Divisor 
2.4 The Euclidean Algorithm 
2.5 The Diophantine Equation 

3 Primes and Their Distribution 
3.1 The Fundamental Theorem of Arithmetic 
3.2 The Sieve of Eratosthenes 
3.3 The Goldbach Conjecture 

4 The Theory of Congruences 
4.1 Carl Friedrich Gauss 
4.2 Basic Properties of Congruence
4.3 Binary and Decimal Representations of Integers 
4.4 Linear Congruences and the Chinese Remainder Theorem

5 Fermat's Theorem 
5.1 Pierre de Fermat 
5.2 Fermat's Little Theorem and Pseudoprimes 
5.3 Wilson's Theorem 
5.4 The Fermat-Kraitchik Factorization Method 

6 Number-Theoretic Functions 
6.1 The Sum and Number of Divisors 
6.2 The Mobius Inversion Formula 
6.3 The Greatest Integer Function 
6.4 An Application to the Calendar 

7 Euler's Generalization of Fermat's Theorem 
7.1 Leonhard Euler 
7.2 Euler's Phi-Function 
7.3 Euler's Theorem 
7.4 Some Properties of the Phi-Function 

8 Primitive Roots and Indices 
8.1 The Order of an Integer Modulo n 
8.2 Primitive Roots for Primes 
8.3 Composite Numbers Having Primitive Roots 
8.4 The Theory of Indices 

9 The Quadratic Reciprocity Law 
9.1 Euler's Criterion 
9.2 The Legendre Symbol and Its Properties 
9.3 Quadratic Reciprocity 
9.4 Quadratic Congruences with Composite Moduli 

10 Introduction to Cryptography 
10.1 From Caesar Cipher to Public Key Cryptography 
10.2 The Knapsack Cryptosystem 
10.3 An Application of Primitive Roots to Cryptography 

11 Numbers of Special Form 
11.1 Marin Mersenne 
11.2 Perfect Numbers 
11.3 Mersenne Primes and Amicable Numbers 
11.4 Fermat Numbers 

12 Certain Nonlinear Diophantine Equations 
12.1 The Equation 
12.2 Fermat's Last Theorem 

13 Representation of Integers as Sums of Squares 
13.1 Joseph Louis Lagrange 
13.2 Sums of Two Squares 
13.3 Sums of More Than Two Squares 

14 Fibonacci Numbers 
14.1 Fibonacci 
14.2 The Fibonacci Sequence 
14.3 Certain Identities Involving Fibonacci Numbers

15 Continued Fractions 
15.1 Srinivasa Ramanujan 
15.2 Finite Continued Fractions 
15.3 Infinite Continued Fractions 
15.4 Farey Fractions 
15.5 Pell's Equation 

16 Some Recent Developments 
16.1 Hardy, Dickson, and Erd�s 
16.2 Primality Testing and Factorization 
16.3 An Application to Factoring: Remote Coin Flipping 
16.4 The Prime Number Theorem and Zeta Function Miscellaneous Problems 

Appendixes 
General References 
Suggested Further Reading 

Tables

David M. Burton

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