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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.

Yuxiao Xie
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  • Why vote to close? What is wrong about this question? Can you please point it out? – Yuxiao Xie Sep 21 '16 at 15:29
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    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 Sep 21 '16 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). – AlvinL Sep 21 '16 at 15:29
  • I think Gentzen-style proofs and generalisations of them are interesting reading as well. – Emil Sep 21 '16 at 15:36
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    Elementary = using tricks, algebraic = using abstract structures, analytic = using analysis. – reuns Sep 21 '16 at 17:40
  • I upvoted this question because I like it because I like the idea of learning how to become good at writing formal proofs of statements describable in Peano arithmetic in the formal system of Peano arithmetic and not just in Zermelo-Fraenkel set theory. It's useful to be able to do it for the sake of being able to do it and because I'm interested in myself eventually self studying and figuring it out because I find the consistency of Zermelo-Fraenkel set theory dubious until I can prove it in number theory. I actually don't know how to write a proof in Peano arithmetic or Zermelo-Fraenkel set – Timothy Jun 11 '19 at 02:30
  • theory but I sense that some day, I'll come up with my own proof system of pure number theory and think maybe it's equivalent to the Formal system of Peano arithmetic and maybe it isn't. I'm one of those people who sometimes doesn't find a proof very intuitive and so might actually be good at starting all over and constructing number theory all by itself without relying on set theory to check its accuracy and rely solely on thinking in number theory to do so. I think the statement that Zermelo-Fraenkel set theory is consistent is equivalent to a statement about number theory that can be proven – Timothy Jun 11 '19 at 02:36
  • in a slightly stronger system than Peano arithmetic such as a system that asserts the statement that all theorems of Peano arithmetic are true. The statement that Peano arithmetic is consistent is describable in Peano arithmetic but not provable in it. On the other hand, the statement that all theorems of Peano arithmetic are true is not even describable in Peano arithmetic. – Timothy Jun 11 '19 at 02:38
  • Does this answer your question? Books on Number Theory for Layman – user1147844 Jun 06 '23 at 07:28

1 Answers1

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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.

Henrique Augusto Souza
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  • Thanks for your answer! What about the prerequisites? So complex analysis, abstract algebra and topology are needed but how deep should I go into these subjects before starting a course in analytic/algebraic number theory? – Yuxiao Xie Sep 22 '16 at 04:47
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    @YuxiaoXie I would say to go cowboy on this: start reading the subject and for every concept you are unfamiliar, go and familiarize yourself with it. But for the books I've cited, just introductory material is enough (an equivalent to a half year course). You'll need only the most elementary stuff from general topology, such as what is a compact set and a induced topology. From abstract algebra, knowing all three basic structures, substructures and quotients and understanding the first isomorphism theorem will take you a long way. And for complex analysis, a bit on holomorphic functions. – Henrique Augusto Souza Sep 22 '16 at 10:50