Elementary Number Theory (2nd Edition)
Quick Informations
Where to buy it: Amazon
Author: David M. Burton
Release date: 1989
Language: English
Personal Progress: I currently read the first eight chapters (more than a half of the book).
Description
This book presents all of the elementary theory of numbers starting from the very basic properties of divisibility to more advanced topics, such as representation of integers into sums of squares or solutions to Pell's equations. Many important topics are mentioned, such as modular arithmetic, prime numbers, primitive roots, the law of quadratic reciprocity, etc. The structure of the book is standard: thirteen chapters, each divided into four of five sections. The book is self-contained.
Review
[I didn't finish reading the book yet.] After the other textbooks I read, I felt like this book was maybe a little bit too easy. I decided to keep reading it, even if this feeling never left. I have this feeling because some entire secions can be dedicated to a very simple theorem or definition, and so I feel like the book is really taking its time to present the subject. In the end, this is not a bad thing because it lets the reader think a lot about these simple ideas while solving the exercises. Overall, I think that it is an excellent textbook to learn the subject, I may be too experienced to read it. This book is perfect for people with little mathematical background, interested in number theory.
Exercises
There are a decent amount of exercises after each section (around fifteen). Some of them are simply computational, and others are more theoretical. For me, the exercises were very easy, but still useful to get familiar with the material. Concerning the computational exercises, most of them can be done really easily using a calculator, or a computer program. It is for this reason that I chose to solve all the exercises in this textbook by hand only. This increases the difficulty because it forces me to use what the author wants me to use, and it lets me practice computations by hand.
There is one big problem with the exercises: whenever an exercise becomes more than just applying a theorem from the previous section, a hint is given directly after the statement of the exercise. This is a problem because it is hard to avoid reading them, but also because some hints give a nearly complete solution of the exercise. I try to not read them whenever I can, but it is difficult. For me, this makes the book way too easy, in an unproductive way.
Prerequisites
This book has no prerequisites. However, I would say that reading a book on proof techniques first would help. For example, it can be useful to read the first two chapters of the book A Transition to Advanced Mathematics first.
Further Readinds
Clearly, this book opens the doors of many subjects. I didn't read books on these subjects yet but I would say that following this book with one on Analytic Number Theory can be a good idea (assuming the reader is familiar with classical analysis). The same is true for Algebraic Number Theory, but I feel like the latter requires more background.