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Elementary Number Theory in Nine Chapters

James J. Tattersall
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michele Intermont
, on

Elementary Number Theory in Nine Chapters does indeed contain the topics I expected to find in an introduction to number theory: divisibility, prime numbers, congruences. There’s even an interesting chapter on generating functions and partitions, and a chapter entitled "Representations," including continued fractions and other specific representations of numbers (as sums of squares, for example).

The book is easy to read. Although talk of writing proofs is not explicit throughout, Tattersall does begin the book by giving the reader an initial foray into the method of induction. Tattersall also includes the sort of historical perspective I want in this subject: some, but not too much.

The feature of the book that pleased me the most is the number of problems that are included. Not only does each section end with a large selection of problems, but each chapter also carries a number of supplementary exercises. There are on the (rough) average, 40 supplementary exercises per chapter, not including the sometimes 20 miscellaneous problems at the end of a chapter.

My disappointment comes in Chapters 4 and 5. Chapter 4, entitled “Perfect and amicable numbers”, introduces lots of “named” numbers: perfect numbers, fermat numbers, amicable numbers, perfect-type numbers. Not to mention deficient numbers, abundant numbers, Euclidean perfect numbers, quasiperfect numbers, superperfect numbers… Yikes! All this comes before the notion of modular arithmetic, which is introduced and explored in Chapter 5. I found myself a bit frustrated and in a rush. The placement of topics should tell me something about importance or logical dependence. Surely dependence is not an issue, and the idea of congruence wins the importance battle here in my opinion. Just as surely, I’ll never remember the definitions of all the specially named numbers.

My other disappointment is less serious. The blurb on the back cover indicates that the reader will find some applications of number theory in the book, most especially cryptography. True, Chapter 7 is entitled “Cryptology,” with sections on alphabetic ciphers, block ciphers, and another on exponential ciphers including RSA. True, whole books are written on the subject of cryptography, and this is not meant to be one of them. I know! But I was still disappointed with the skimpiness of the treatment of the subject.

Michele Intermont is an Associate Professor at Kalamazoo College in Kalamazoo, Michigan. Her interests include swimming, biking, and topology, many times in that order!

Preface page ix

1 The intriguing natural numbers

1.1 Polygonal numbers 1

1.2 Sequences of natural numbers 23

1.3 The principle of mathematical induction 40

1.4 Miscellaneous exercises 43

1.5 Supplementary exercises 50

2 Divisibility

2.1 The division algorithm 55

2.2 The greatest common divisor 64

2.3 The Euclidean algorithm 70

2.4 Pythagorean triples 76

2.5 Miscellaneous exercises 81

2.6 Supplementary exercises 84

3 Prime numbers

3.1 Euclid on primes 87

3.2 Number theoretic functions 94

3.3 Multiplicative functions 103

3.4 Factoring 108

3.5 The greatest integer function 112

3.6 Primes revisited 115

3.7 Miscellaneous exercises 129

3.8 Supplementary exercises 133

4 Perfect and amicable numbers

4.1 Perfect numbers 136

4.2 Fermat numbers 145

4.3 Amicable numbers 147

4.4 Perfect-type numbers 150

4.5 Supplementary exercises 159

5 Modular arithmetic

5.1 Congruence 161

5.2 Divisibility criteria 169

5.3 Euler’s phi-function 173

5.4 Conditional linear congruences 181

5.5 Miscellaneous exercises 190

5.6 Supplementary exercises 193

6 Congruences of higher degree

6.1 Polynomial congruences 196

6.2 Quadratic congruences 200

6.3 Primitive roots 212

6.4 Miscellaneous exercises 222

6.5 Supplementary exercises 223

7 Cryptology

7.1 Monoalphabetic ciphers 226

7.2 Polyalphabetic ciphers 235

7.3 Knapsack and block ciphers 245

7.4 Exponential ciphers 250

7.5 Supplementary exercises 255

8 Representations

8.1 Sums of squares 258

8.2 Pell’s equation 274

8.3 Binary quadratic forms 280

8.4 Finite continued fractions 283

8.5 Infinite continued fractions 291

8.6 p-Adic analysis 298

8.7 Supplementary exercises 302

9 Partitions

9.1 Generating functions 304

9.2 Partitions 306

9.3 Pentagonal Number Theorem 311

9.4 Supplementary exercises 324


T.1 List of symbols used 326

T.2 Primes less than 10 000 329

T.3 The values of τ(n), σ(n), φ(n), μ(n), ω(n), and Ω(n) for natural numbers less than or equal to 100 333

Answers to selected exercises 336


Mathematics (general) 411

History (general) 412

Chapter references 413