Since summer comes with a lot of spare time, I’ve decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really caught my eye, but I have no prior training in it.
I’ve decided to conduct my studying effort in a library; I prefer real books to virtual ones, but as I’m not allowed to browse through the books on my own, I have to know beforehand what I’m looking for and this is where I’m kind of lost. I’m not really sure where to start.
Basically I wanna know about the following:
What are the prerequisites?
(I’m currently trained in Linear Algebra, Calculus, Complex Analysis – all on an undergraduate level )
Can you recommend some reading materials?
I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested in Algebraic Geometry and Number Theory, what would be a textbook that has everything an undergraduate should know about Number Theory before moving on. He told me that A Classical Introduction to Modern Number Theory by Kenneth F. Ireland and Michael Rosen is the perfect choice. He also mentioned that I should definitely study chapters 1-8,10-13 and 17. Another book that he mentioned was A Friendly Introduction to Number Theory by Joseph H. Silverman. He emphasized though that this book is clearly an introduction whereas the previous one gives you all the tools you need in order to study many things that are connected to Number Theory. I hope that this helped you!
In my opinion Hardy &Wright’s book on Number Theory is not the best possible book for someone “who has no prior training in Number Theory”, I would suggest the following books.
Elementary Number theory by David M. Burton.
Number Theory A Historical Approach by John H. Watkins
Higher Arithmetic by H. Davenport
All the books are well-written. I think that if you are a beginner, and if you are interested in the historical aspects of Number Theory as well, you may first look at the second book. Although Burton’s books also have some historical background in each chapter. I would suggest reading Davenport’s books a bit later when you have a fair grasp of the subject.
Also I suggest you to look at the suggestions given at this post.
One of the best is An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery.
I would recommend An Introduction to the Theory of Numbers By G.H. Hardy and E.M. Wright .
Dover publishes many number theory titles. At \$10-\$15 each they’re a bargain – no need even to look for the Amazon discount. You can get several and jump back and forth among them to get different perspectives on each topic. You can write yourself notes in the margins. Take them to the library to read.
This is a standard old undergraduate text:
Elementary Number Theory: Second Edition
Way off the beaten path, but fun, with a rarely encountered (in elementary texts) proof of quadratic reciprocity:
An Adventurer’s Guide to Number Theory
I can certainly recommend Elementary Number Theory by Gareth A Jones et al. It will get you started and then you can move onto more advanced texts. It’s a very short book (about 300 pages) which means you can easily read through the whole text-a very good choice for self study. As for pre-requisites, a good grasp of algebra will probably do.