# Pathway to number theory?

My questions come down to these two:

1. What are the major branches of number theory?
2. 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.

• Why vote to close? What is wrong about this question? Can you please point it out? Commented Sep 21, 2016 at 15:29
• Not sure if this covers what you said but I have found a few jokes in it which was enheartening while reading: Abstract Algebra, Pierre Antoine Grillet
– Emil
Commented Sep 21, 2016 at 15:29
• Get acquainted with propositional calculus (Robert S.Wolfe's "A tour through mathematical logic", for instance) and some preliminary understanding on algebraic structures (which you seem to be getting). Commented Sep 21, 2016 at 15:29
• I think Gentzen-style proofs and generalisations of them are interesting reading as well.
– Emil
Commented Sep 21, 2016 at 15:36
• Elementary = using tricks, algebraic = using abstract structures, analytic = using analysis. Commented Sep 21, 2016 at 17:40