My questions come down to these two:
- What are the major branches of number theory?
- What is a recommended pathway to these branches of number theory from only elementary mathematics (those covered in the high school curriculum)? In particular, what do I need to study? What are some texts that suit for this purpose?
Here's my background. I'm a high school student who started to study number theory several months ago and quickly got fascinated by this beautiful subject. So I decided to delve deep into it. But then I realized that it's such a huge subject with so many branches to study, and what's more, most of these more advanced topics exploit tools from higher mathematics, which I know little about. With so many things to learn (number theory itself and so many prerequisites) I don't know where to start. Therefore I ask this question, seeking a self-study pathway so that I can make a study plan.
I'm more "mathematically mature" than ordinary high school students because first, I have been preparing for (high school level) mathematical competitions and second, I was exposed to higher mathematics already. I know some basic concepts from real analysis, linear algebra, combinatorics etc. (But that doesn't mean I've learned those.) So please recommend serious texts. By the way, I'm reading Hardy & Wright's An Introduction to the Theory of Numbers and Thomas Hungerford's Algebra (GTM 73) for a foundation in abstract algebra.
I hope this question will not be closed. I think a lot of people (like freshmen) can benefit from such a pathway. But for me, I don't have anyone to mentor me so I really need a detailed pathway, from which I can learn what exactly I need to do. This is really important to me and I will really appreciate your help.
Best Answer
Elementary number theory, i.e., modular congruence, linear Diophantine equations, quadratic residues and quadratic reciprocity can be easily studied without experience in higher algebra or calculus, and there a probably several books on the topic accessible to you (in almost every mother tongue). Really, it is a question of taste. One of my favorites is Elementary Number Theory, by Gareth and Josephine Jones. Also, it will cover basic and central concepts of number theory you'll need in any advanced study.
From there, number theory breaks in two major branches (with a great deal of overlap between them, and, of course, not comprehensive branches): analytic and algebraic number theory. For analytic, some calculus may come in hand. As for algebraic, basic higher algebra will be expected.
The standard reference of analytic number theory is Apostol's Introduction to Analytic Number Theory. How much calculus you'll need depends on how deep will you go (it can be single variable, multi variable, complex variable, you'll may also need some general topology, who knows?).
I've never read something specific about algebraic number theory, but the Internet seems to recommend Rosen's & Ireland's A Classical Introduction to Modern Number Theory. From the summary, it appears to cover the basics of algebraic number theory (also, with a lot of overlap with elementary stuff). I've had a good experience with Serre's A Course in Arithmetic, which covers both analytic and algebraic aspects, though it's a very hard book to digest (also, the french original is superior).
From there, you'll probably already have very specific interests from which you will be able to get better references. And for the calculus, algebra and topology, you can find introductory references here on the site. Expect to need at least some knowledge of groups, rings and fields, derivatives, infinite series, topology and complex variables to delve in the most advanced (yet central) areas.