Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- 1 The fundamental theorem of arithmetic
- 2 Modular addition and Euler's ɸ function
- 3 Modular multiplication
- 4 Quadratic residues
- 5 The equation xn + yn = zn, for n = 2, 3, 4
- 6 Sums of squares
- 7 Partitions
- 8 Quadratic forms
- 9 Geometry of numbers
- 10 Continued fractions
- 11 Approximation of irrationals by rationals
- Bibliography
- Index
Summary
The construction of the Pathway
Have you attended a mathematics lecture, followed each step of the argument, and yet at the end felt that you did not understand what it was about? Have you read a proof of a theorem in a book and felt the same? If so, you have experienced a feeling common to most mathematicians.
This book on number theory has been put together by keeping a record of how I actually resolved the blocks which I encountered as I read a number of standard texts. Time and again, it was the exploration of special cases which illuminated the generalities for me. This collection of explorations was then organised into a sequence in such a way that the ‘pathway’ would climb towards the standard theorems which occur here as problems for the student at the end of each section.
The motivation for assembling the Pathway was a college need to mount a course for which lectures would not be given. If the Pathway is more successful than some other books or undergraduate lecture courses in number theory, it is because it follows more closely than usual the natural process of discovery, and puts logic in its proper place. The purpose of rigour', said Hadamard, ‘is to legitimate the conquests of the intuition, and it has never had any other purpose'. Formality, abstraction and generality have an essential place in the completion of any piece of mathematics, but their role in discovery is varied. In the Pathway, the introduction to each new idea is as informal and as specific as I could make it. There are a few remarks about foundations in the notes, but the statement of the Peano axioms postdates almost all of the number theory in the book and is not given here.
In the selection of material, the needs of future teachers have been kept in mind. The theme of sums of squares links chapters 4, 5, 6, 8, 9 and 10. Arithmetic functions and the distribution of primes were thought to offer less connection with school work.
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- Information
- A Pathway Into Number Theory , pp. xiii - xviPublisher: Cambridge University PressPrint publication year: 1996