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Publisher:

Cambridge University Press

Publication Date:

1997

Number of Pages:

278

Format:

Paperback

Edition:

2

Price:

53.00

ISBN:

0521575400

Category:

Textbook

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

[Reviewed by , on ]

Allen Stenger

01/19/2009

This book studies number theory the old-fashioned way — by looking at lots of numbers. Most of the sections start out with a big table of numbers. For example, to start the discussion of the Diophantine equation x^{2} + y^{2} = z^{2}, we have a whole page listing the squares and using that to form the numbers that are sums of two squares. We are then invited to circle all the squares in the table and look for a pattern.

The style of the book is similar to Pólya and Szegö’s Problems and Theorems in Analysis. The book consists of many long sequences of problems, each leading to a notable result, and there are very brief solutions in the back of each chapter. The Burn book is much easier that Polya & Szego, partly because it does not go very deep and partly because there are many more steps. The pace seems very slow to someone who is already familiar with the material, but I think it does give a good feel for how research is really done, with lots of experimentation and wandering around.

The book makes heavier use of drawings and graphs than is typical in beginning number theory books. The topics are typical of beginning number theory books, with a few surprises such as quadratic forms and quite a lot on the geometry of numbers.

*Pathway* is advertised as requiring only a high-school background. This is a slight exaggeration as it makes some use of abstract algebra. It also makes some use of complex numbers and quaternions.

The book’s first edition was in 1982, but the book is what we would today call “inquiry-based.” I compared it to the 2007 volume Number Theory Through Inquiry by Marshall, Odell, and Starbird. The latter book is much more conventional; in fact to some extent it looks like a plain number theory book with definitions, theorems, and examples, but with the proofs left out. The latter authors are also very interested in pedagogy and they have a lot of tips and exercises to see if you are learning the material; Burn is only interested in numbers.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Preface to the second edition

Introduction

1. The fundamental theorem of arithmetic

Division algorithm

Greatest common divisor and Euclidean algorithm

Unique factorisation into primes

Infinity of primes

Mersenne primes

Summary

Historical note

Notes and answers

2. Modular addition and Euler’s φ function

Congruence classes and the Chinese remainder theorem

The groups (**Z**_{n}, +) and their generators

Euler’s φ function

Summing Euler’s function over divisors

Summary

Historical note

Notes and answers

3. Modular multiplication

Fermat’s theorem

Wilson’s theorem

Linear congruences

Fermat-Euler theorem

Simultaneous linear congruences

Lagrange’s theorem for polynomials

Primitive roots

Chevalley’s theorem

RSA codes

Summary

Historical note

Notes and answers

4. Quadratic residues

Quadratic residues and the Legendre symbol

Gauss’ lemma

Law of quadratic reciprocity

Summary

Historical note

Notes and answers

5. The equation x^{n} + y^{n} = z^{n}, for n = 2, 3, 4

The equation x^{2} + y^{2} = z^{2}

The equation x^{4} + y^{4} = z^{4}

The equation x^{2} + y^{2} + z^{2} = t^{2}

The equation x^{3} + y^{3} = z^{3}

Historical note

Notes and answers

6. Sums of squares

Sums of two squares

Sums of four squares

Sums of three squares

Triangular numbers

Historical note

Notes and answers

7. Partitions

Ferrers’ graphs

Generating functions

Euler’s theorem

Summary

Historical note

Notes and answers

8. Quadratic forms

Unimodular transformations

Equivalent quadratic forms

Discriminant

Proper representation

Reduced forms

Automorphs of definite quadratic forms

Summary

Historical note

Notes and answers

9. Geometry of numbers

Subgroups of a square lattice

Minkowski’s theorem in two dimensions

Subgroups of a cubic lattice

Minkowski’s theorem in three dimensions

Legendre’s theorem on ax^{2} + by^{2} + cz^{2} = 0

Summary

Historical note

Notes and answers

10. Continued fractions

Irrational square roots

Convergence

Purely periodic continued fractions

Pell’s equation

Lagrange’s theorem on quadratic irrationals

Automorphs of the indefinite form ax^{2} – by^{2}

Summary

Historical note

Notes and answers

11. Approximation of irrationals by rationals

Naive approach

Farey sequences

Hurwitz’ theorem

Liouville’s theorem

Summary

Historical note

Notes and answers

Bibliography

Index

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