Good Number Theory books and introductions

Eliot Rosewater

As part of my degree I did some elementary number theory way back in first year. I found it very interesting. Im just wondering could anyone recommend some good number theory books? Its been a while since I studied it so Ideally something that would start from the beginning. I appreciate any suggestions

hivizman

A reasonable introduction is Jones & Jones Elementary Number Theory (Springer), which is pitched towards undergraduate courses. A bit more advanced is T M Apostol's Introduction to Analytic Number Theory - this is perhaps more a final year undergraduate/Master's level book, and the later chapters require knowledge of some complex analysis. The classic textbook, which I used when I studied number theory a long time ago, is Hardy & Wright An Introduction to the Theory of Numbers (OUP) - this recently had a new edition, but it may be a bit dated in comparison with the other books mentioned.

Fremen

+1 to hivizman's recommendations.

Apostol doesn't require very much knowledge at all. A little mathematical "sophistication" is all you need. A very good leaving cert student could probably handle it, except for the last few chapters.
Hardy and wright is the standard reference, but it's a bit of a tome. It would be very tough to read cover to cover. However, the proofs are all clear and well-written.

There are three main branches of number theory: analytic, algebraic and elementary. Analytic uses calculus, and complex analysis in particular. Algebraic uses tools from algebra like rings, groups and fields. Elementary uses more basic arguments, but often in a more complicated way. Erdos' and Selberg's proof of the prime number theorem is elementary but it is by no means easy.

Eliot Rosewater

Thanks for the recommendations (a year later)!

I recently did a project on number theory - on Mordell's equation - and I'm starting a number theory course, so I intend to buy a book soon money permitting. My project was rushed and I didn't enjoy it at all, so now I want a more solid foundation and to able to appreciate the subject. I read A Mathematician's Apology by Hardy and really enjoyed it, so that tips the decision in his favour. The chapter list in the Number Theory book is extensive, and there are a lot of topics covered there that I didn't see covered in any of the other textbooks I took out in the course of my project work (H.E. Rose's book, for instance).

sponsoredwalk

I only have a superficial understanding of number theory but am going to
start reading Invitation to Number Theory in the next week or so (whenever
it arrives) as it's short, part of a great series of books & just to get a proper
feel for the subject. Then I'm going to do the Jones ENT book spoken about
along with Elementary Number Theory - Rosen, just to get that kind of book
out of the way. A Primer of Analytic Number Theory along with Apostol is
the top of the summit

Fremen

What material will you be covering in your number theory course?

Number theory gets really weird as you get more advanced in the subject. At first glance, it's surprising that complex analysis should be able to tell you something about the prime numbers (via the Riemann zeta function). But it gets much wierder, incorporating ideas from non-Euclidean geometry, abstract algebra, and topology.

You should be aware that there aren't that many number theorists in Ireland - Sander Zweigers is the only one I could name off the top of my head. If you really get into it, you may eventually have to leave Ireland to study it.

LeixlipRed

Pat McCarthy in Maynooth too. Doesn't publish and would count combinatorics as his main love but has had a few Phd students in Number Theory.

Eliot Rosewater

I'm not getting into number theory with any career/research plan in mind. The subject and the questions asked in it merely look intriguing!

The Number theory course is pretty bare, as it's only one third of a module "Introduction to Discrete Mathematics". The lecturer we have for it is extremely poor: he's not going to teach it well and he's not going to make it interesting, which is another reason I've dug up this thread. The number theory topics we'll be covering are:

• Use Euclid's algorithm to find the largest common divisor of 2 numbers (we already did this in first year).
• Solve linear Diophantic equations.
• Prove and apply properties of multiplicative functions such as the Euler phi-function.
• Check when a number is a quadratic residue modulo n.

The Hardy & Wright book is on elementary number theory. So, when I've done a bit of that, would I just get an introduction to analytic number theory to get into that? Judging from the contents of the book sponsoredwalk linked to, there doesn't seem to be much of a crossover??

Fremen

To be honest, I find the elementary stuff a bit boring. If you're competent at calculus, you could just jump straight into the analytic side with Apostol's book. It would help you see the "bigger picture" of number theory, but wouldn't necessarily get you a good grade in your class.

cartier

Fremen wrote: »

You should be aware that there aren't that many number theorists in Ireland - Sander Zweigers is the only one I could name off the top of my head. If you really get into it, you may eventually have to leave Ireland to study it.

um, what about this guy?

http://mathsci.ucd.ie/~osburn/

Belfast

The Theory of Numbers by R. D. Carmichael
http://www.gutenberg.org/ebooks/13693

Essays on the Theory of Numbers by Richard Dedekind
http://www.gutenberg.org/ebooks/21016

Eliot Rosewater

Just an update for anyone who finds this thread - the book An Introduction to the Theory of Numbers by GH Hardy and EM Wright is, from my experience of the first 4 chapters so far, absolutely brilliant. It's not easy to read, in so far as it's very dense and every few sentences require thought and exercise, but the concepts are presented very clearly, and the proofs they give are really lovely.

There's been a lot of "ahhh" moments, when some really nice proof is given, or when some concept one was already familiar with - GCD, for example - is presented in a new way that makes it far clearer and far simpler to understand.

You know a maths textbook is good when you find it as enjoyable at 9 o'clock at night as Grand Theft Auto!

sponsoredwalk

I'm still far too busy to do a proper number theory book but I wasn't happy
relying on that Rosen book above. I did some browsing & came across two
absolute gems:

Number Theory: Structures, Examples & Problems
Number theory: an introduction to mathematics

The first book is an olympiad book with problems in each section but at the
beginning of each section there is theory with theorems, proofs & Examples.

The second book is one after my own heart, a book full of theorems w/
proofs with no end of section questions at all, just pure theory. Looks pretty
advanced in the second half, these two might interest you.

wallst

Do you guys know anyone who would be prepared to give a 3-4 hour one-off grind to an undergraduate who is struggling at Number Theory? Student can travel anywhere for it. Studying an elementary number theory course only and needs a 50% grade on exam.

TheBody

wallst wrote: »

Do you guys know anyone who would be prepared to give a 3-4 hour one-off grind to an undergraduate who is struggling at Number Theory? Student can travel anywhere for it. Studying an elementary number theory course only and needs a 50% grade on exam.

Try having a look or posting in the grinds tread sticky at the top of the maths forum.

http://www.boards.ie/vbulletin/showthread.php?t=2055800688&page=11