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.