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Number Theory: A Lively Introduction with Proofs, Applications, and Stories

James E. Pommersheim, Tim K. Marks, and Erica L. Flapan
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Miklós Bóna
, on

This is a very accessible and well-written book whose best use is as a textbook for a course designed for non-majors or for students who are preparing to be mathematics teachers in high school. While the material that is covered is serious and the coverage itself is rigorous, the topics are not quite advanced enough nor diverse enough for the usual course taken by mathematics majors. No calculus background is required, so even a talented and dedicated high-school student can attempt to read the book.

Each chapter starts by explaining the historical context of the results to be shown, and by the introduction of an important historical figure. There is often a funny fictional conversation between two people from very different centuries. This puts the reader in the mood for the upcoming topics.

The introductory part is long. The fact that there are infinitely many primes, which is the first theorem in many number theory classes, is proved on page 105. This is because the first two chapters teach the non-mathematics student how to prove statements, by induction or by other means.

Chapters 3–9 are still quite basic, and are mostly about divisibility issues and modular arithmetic. Many mathematics majors see this material in a Transition to Advanced Mathematics course, so the first half of the book would not be new for them.

Chapter 10–15 are on more advanced topics, such as primality testing, quadratic residues, and Gaussian integers. These chapters are independent, so the instructor is free to choose any subset of them. Multiplicative algebraic number theory is the dominant overarching theme.

There are plenty of exercises, and the book is a pleasure to read. If you have the right audience for it, you will enjoy teaching from this book as well.

Miklós Bóna is Professor of Mathematics at the University of Florida.


To the Student.

To the Instructor.


0. Prologue.

1. Numbers, Rational and Irrational.

(Historical figures: Pythagoras and Hypatia).

1.1 Numbers and the Greeks.

1.2 Numbers you know.

1.3 A First Look at Proofs.

1.4 Irrationality of he square root of 2.

1.5 Using Quantifiers.

2. Mathematical Induction.

(Historical figure: Noether).

2.1.The Principle of Mathematical Induction.

2.2 Strong Induction and the Well Ordering Principle.

2.3 The Fibonacci Sequence and the Golden Ratio.

2.4 The Legend of the Golden Ratio.

3. Divisibility and Primes.

(Historical figure: Eratosthenes).

3.1 Basic Properties of Divisibility.

3.2 Prime and Composite Numbers.

3.3 Patterns in the Primes.

3.4 Common Divisors and Common Multiples.

3.5 The Division Theorem.

3.6 Applications of gcd and lcm.

4.The Euclidean Algorithm.

(Historical figure: Euclid).

4.1 The Euclidean Algorithm.

4.2 Finding the Greatest Common Divisor.

4.3 A Greeker Argument that the square root of 2 is Irrational.

5. Linear Diophantine Equations.

(Historical figure: Diophantus).

5.1 The Equation aX + bY = 1.

5.2 Using the Euclidean Algorithm to Find a Solution.

5.3 The Diophantine Equation aX + bY = n.

5.4 Finding All Solutions to a Linear Diophantine Equation.

6. The Fundamental Theorem of Arithmetic.

(Historical figure: Mersenne).

6.1 The Fundamental Theorem.

6.2 Consequences of the Fundamental Theorem.

7. Modular Arithmetic.

(Historical figure: Gauss).

7.1 Congruence modulo n.

7.2 Arithmetic with Congruences.

7.3 Check Digit Schemes.

7.4 The Chinese Remainder Theorem.

7.5 The Gregorian Calendar.

7.6 The Mayan Calendar.

8. Modular Number Systems.

(Historical figure: Turing).

8.1 The Number System Zn: an Informal View.

8.2 The Number System Zn: Definition and Basic Properties.

8.3 Multiplicative Inverses in Zn.

8.4 Elementary Cryptography.

8.5 Encryption Using Modular Multiplication.

9. Exponents Modulo n.

(Historical figure: Fermat).

9.1 Fermat's Little Theorem.

9.2 Reduced Residues and the Euler phi-function.

9.3 Euler's Theorem.

9.4 Exponentiation Ciphers with a Prime modulus.

9.5 The RSA Encryption Algorithm.

10. Primitive Roots.

(Historical figure: Lagrange).

10.1 Zn.

10.2 Solving Polynomial Equations in Zn.

10.3 Primitive Roots.

10.4 Applications of Primitive Roots.

11. Quadratic Residues.

(Historical figure: Eisenstein)

11.1 Squares Modulo n

11.2 Euler's Identity and the Quadratic Character of -1

11.3 The Law of Quadratic Reciprocity

11.4 Gauss's Lemma

11.5 Quadratic Residues and Lattice Points.

11.6 The Proof of Quadratic Reciprocity.

12. Primality Testing.

(Historical figure: Erdös).

12.1 Primality testing.

12.2 Continued Consideration of Charmichael Numbers.

12.3 The Miller-Rabin Primality test.

12.4 Two Special Polynomial Equations in Zp.

12.5 Proof that Millar-Rabin is Effective.

12.6 Prime Certificates.

12.7 The AKS Deterministic Primality Test.

13. Gaussian Integers.

(Historical figure: Euler).

13.1 Definition of Gaussian Integers

13.2 Divisibility and Primes in Z[i].

13.3 The Division Theorem for the Gaussian Integers.

13.4 Unique Factorization in Z[i].

13.5 Gaussian Primes.

13.6 Fermat's Two Squares Theorem.

14. Continued Fractions.

(Historical figure: Ramanujan).

14.1 Expressing Rational Numbers as Continued Fractions.

14.2 Expressing Irrational Numbers as Continued Fractions.

14.3 Approximating Irrational Numbers Using Continued Fractions.

14.4 Proving that Convergents are Fantastic Approximations.

15. Some Nonlinear Diophantine Equations.

(Historical figure: Germain).

15.1 Pell's Equation

15.2 Fermat's Last Theorem

15.3 Proof of Fermat's Last Theorem for n = 4.

15.4 Germain's Contributions to Fermat's Last Theorem

15.5 A Geometric look at the Equation x4 + y4 = z2.

Appendix: Axioms of Number Theory.

A.1 What is a Number System?

A.2 Order Properties of the Integers.

A.3 Building Results From Our Axioms.

A.4 The Principle of Mathematical Induction.