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Elementary Number Theory

David M. Burton
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John Cullinan
, on

This book is primarily a classical introduction to the subject, with a few modern applications. In the preface, Burton describes how the book's sixteen chapters divide naturally into two parts. The first nine chapters give a detailed introduction to the basics of elementary number theory: Euclidean algorithm, Sieve of Eratosthenes, Chinese remainder theorem, Fermat's and Wilson's theorems, Möbius inversion, Euler's phi-function, and quadratic reciprocity. Chapters 10-16 are independent of one another and cover a variety of topics, such as Cryptography, Nonlinear Diophantine Equations, Continued Fractions, and the Riemann zeta function.

The book is full of examples and proofs are carried out in detail. The exercises are plentiful and range from routine to moderately difficult. Burton writes clearly and goes to great pains not to lose the reader during the more intricate arguments; this is most apparent in his treatment of quadratic reciprocity. He builds up the theory slowly through many examples and short theorems. By the time he arrives at Gauss' Lemma (the main tool in his proof of quadratic reciprocity), the reader has become an expert on the various properties of the Legendre symbol.

What makes this book especially enjoyable is the historical background provided for each new topic. These accounts are short (typically 1-2 pages), frequent, and blend seamlessly with the mathematics. My favorite examples come from chapter 15 — Continued Fractions. Burton begins with a section on Ramanujan's life and his contributions to the subject. Next he develops properties of finite and infinite continued fractions, complete with historical anecdotes. He concludes the chapter with a detailed history and exposition of Pell's equation. His approach works very well; he piques the reader's curiosity with the beauty of Ramanujan's formulæ, then hints of the connection between periodic continued fractions and Pell's equation. This sets the tone for the rest of the chapter and provides clear insight into a subject which can often seem mysterious.

The mathematical content of the last seven chapters is not as deep as that of the first nine, which is to be expected due to their logical independence. In the preface, Burton states that the second half of the book is well-suited for student presentations and extra credit projects. I agree with this, but I would have liked a bibliography, or suggestions for further reading, at the end of each chapter (as in Gallian's Contemporary Abstract Algebra). This would be especially useful for anyone using the book for independent study or as a resource for a class presentation.

Overall I think this is a great book, appropriate for a number of different courses in number theory. His effort to provide the reader with the stories behind the mathematics is greatly appreciated.

John Cullinan is Visiting Assistant Professor of Mathematics at Colby College.




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

2.1 The Division Algorithm

2.2 The Greatest Common Divisor

2.3 The Euclidean Algorithm

2.4 The Diophantine Equation ax + by = c

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 Möbius 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 x2 + y2 = z2

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 Pell's Equation

16 Some Twentieth-Century Developments

16.1 Hardy, Dickson, and Erdös

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


General References
Suggested Further Reading
Answers to Selected Problems