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A Pathway Into Number Theory

R. P. Burn
Cambridge University Press
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[Reviewed by
Allen Stenger
, on

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 x2 + y2 = z2, 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.


Preface to the second edition


1. The fundamental theorem of arithmetic
Division algorithm
Greatest common divisor and Euclidean algorithm
Unique factorisation into primes
Infinity of primes
Mersenne primes
Historical note
Notes and answers

2. Modular addition and Euler’s φ function
Congruence classes and the Chinese remainder theorem
The groups (Zn, +) and their generators
Euler’s φ function
Summing Euler’s function over divisors
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
Historical note
Notes and answers

4. Quadratic residues
Quadratic residues and the Legendre symbol
Gauss’ lemma
Law of quadratic reciprocity
Historical note
Notes and answers

5. The equation xn + yn = zn, for n = 2, 3, 4
The equation x2 + y2 = z2
The equation x4 + y4 = z4
The equation x2 + y2 + z2 = t2
The equation x3 + y3 = z3
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
Historical note
Notes and answers

8. Quadratic forms
Unimodular transformations
Equivalent quadratic forms
Proper representation
Reduced forms
Automorphs of definite quadratic forms
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 ax2 + by2 + cz2 = 0
Historical note
Notes and answers

10. Continued fractions
Irrational square roots
Purely periodic continued fractions
Pell’s equation
Lagrange’s theorem on quadratic irrationals
Automorphs of the indefinite form ax2 – by2
Historical note
Notes and answers

11. Approximation of irrationals by rationals
Naive approach
Farey sequences
Hurwitz’ theorem
Liouville’s theorem
Historical note
Notes and answers