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Which is the single best book for Number Theory that everyone who loves Mathematics should read?

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@Charles Stewart: What does the big-list tag mean? – Prasoon Saurav Jul 21 '10 at 13:48
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@Prasoon: It's from Math Overflow: it means that there isn't one right answer your your question, but instead you expect lots of alternative answers. – Charles Stewart Jul 21 '10 at 13:59
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@Prasoon: Those types of questions are typically also community wiki, for the same reason. – Larry Wang Jul 21 '10 at 14:36
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I think "big-list" is a very useful tag. – Noah Snyder Jul 21 '10 at 16:49
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@quanta But what was the problem with the link to the Wikipedia article about number theory? – Adrián Barquero Apr 26 '11 at 16:42
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25 Answers

up vote 51 down vote accepted

A Classical Introduction to Modern Number Theory by Ireland and Rosen hands down!

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+1 Their A Classical Introduction to Modern Number Theory is an excellent treatise on number theory that covers a lot of material in an intuitive and friendly, yet rigorous, presentation. The only bad thing is that you cannot skip around. – Daniel Trebbien Sep 26 '10 at 15:35
It's VERY hard to argue with anyone that picks this awesome text-it's certainly the best book for strong undergraduate mathematics majors.But there are just SO many good textbooks on this ancient and critical subject,I don't think there's a unique answer to it. – Mathemagician1234 Sep 12 '12 at 7:06
up vote 36 down vote

I would still stick with Hardy and Wright, even if it is quite old.

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A new edition came out just a few years back, and a lot of effort was put into bringing it up to date. Also, the new edition has an index, at long last! – Gerry Myerson Apr 26 '11 at 13:08
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Serre's "A course in Arithmetic" is pretty phenomenal.

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up vote 20 down vote

I like Niven and Zuckerman, Introduction to the Theory of Numbers.

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The most recent edition, which is Niven, Zuckerman, and Montgomery, is even better than the earlier editions, which were very good. – Gerry Myerson Apr 26 '11 at 13:06
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This book is great. Back in college, I heard someone mention a warning he had heard: "Be very wary of reading this book; it may turn you into a number theorist". The guy who said this to me did read the book, and turned into a number theorist. – ShreevatsaR Sep 5 '12 at 2:30
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I recommend Primes of the Form x2 + ny2, by David Cox. The question of which primes can be written as the sum of two squares was settled by Euler. The more general question turns out to be much harder, and leads you to more advanced techniques in number theory like class field theory and elliptic curves with complex multiplication.

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up vote 11 down vote

A concise introduction to the theory of numbers by Alan Baker (1970 Fields medalist) covers a lot of ground in less than 100 pages, and does so in a fluid way that never feels rushed. I love this little book.

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up vote 10 down vote

There are many books on this list that I'm a fan of, but I'd have to go with Neukirch's Algebraic Number Theory. Great style, great selection of topics.

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I totally agree. It's both accessible and up to date. But I'm not sure the book could be read as an introduction. – Joel Cohen May 8 '11 at 3:47
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Apostol, Introduction to Analytic Number Theory. I think it' very well written, I got a lot out of it from self-study.

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up vote 9 down vote

It depends on the level.

For an undergraduate interested in algebraic number theory, I would strongly suggest (parts of) Serre's Cours d'arithmetique and also Samuel's Théorie algébriques des nombres.

For a graduate student aiming at a future of research work in number theory, Cassels & Fröhlich is a must.

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Well, how about the Jurgen Neukirch? – awllower Mar 1 '11 at 11:38
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Is Cassels and Frohlich still a must? I had the impression that Neukirch, or Milne's notes jmilne.org/math/CourseNotes/cft.html were adequate substitutes, and perhaps more readable. – David Speyer Jun 29 '11 at 3:11
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Problems in Algebraic Number Theory is written in a style I'd like to see in more textbooks

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up vote 8 down vote

One of my colleagues, a number theorist, recommended the little book by van den eynden for beginners. my favorite is by trygve nagell. (I am a geometer.) One of my friends, preparing for a PhD in arithmetic geometry?, started with the one recommended by Barry, Basic number theory. As I recall it's for people who can handle Haar measure popping up on the first page of a "basic" book on number theory.

I also recommend Gauss's Disquisitiones Arithmeticae.

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Dear Roy, Your memory of Weil's book is correct. It was a bit of a shock to me the first time I opened it (and the shock has never entirely worn off). – Matt E Dec 31 '10 at 4:31
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Yes, Matt, me too, and it could explain why I have no memory at all of page 2. – roy smith Jan 5 '11 at 20:03
+1 for the Disquisitiones Arithmeticae. – awllower Mar 1 '11 at 11:41
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Basic Number Theory by Andre Weil. It's hard going and mind-blowing.

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It is already mentioned. – awllower Mar 1 '11 at 11:51
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Yes, by me. And by someone else referring to my comment...I don't understand why you come along three months later and post this. – Barry Smith Apr 4 '11 at 18:02
Excuse moi? What are you referring to? – awllower Apr 5 '11 at 6:04
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For you are the first, I agree; for the three months later, well, it's because I forgot this website at that time, and recently I came up with it, so... – awllower Apr 9 '11 at 2:37
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This is so far the funniest conversation I ever saw on MSE.(LoL) – Tomarinator May 18 '12 at 11:34
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up vote 6 down vote

My favorite is Elementary Number Theory by Rosen, which combines computer programming with number theory, and is accessible at a high school level.

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+1. It's not my favorite,but for sheer fun readability and scholarship that can inspire the raw beginner, Rosen's very hard to beat! – Mathemagician1234 Sep 12 '12 at 7:10
up vote 5 down vote

Elementary Number Theory - by David M. Burton if you want it somewhere halfway between fast and slow.

link

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its my favorite as it is written in a simple language and is perfect for self studying + can be easily understood by a high schooler. – shrey Dec 12 '12 at 11:53
up vote 5 down vote

One book I think everyone should see is the one by Joe Roberts, Elementary Number Theory : A Problem Oriented Approach. First reason: the first third of the book is just problems, then the rest of the book is solutions. Second reason: the whole book is done in calligraphy.

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Not a really bad book, apart from the calligraphy, which is a truly terrible idea. – André Nicolas Sep 5 '12 at 2:03
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For a highly motivated account of analytic number theory, I'd recommend Harold Davenport's Multiplicative Number Theory.

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up vote 4 down vote

Stewart&Tall's "Algebraic Number Theory" is great.

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up vote 4 down vote

Manin and Panchishkin's Introduction to Modern Number Theory

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broken link, as far as i can tell – Sam Lichtenstein Sep 26 '10 at 17:31
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works fine when I try – Who Sep 27 '10 at 18:00
in my case, it is blocked by the software for the bad reputation of this site, maybe you should try another way of putting on this book, thanks. – awllower Mar 1 '11 at 11:50
up vote 4 down vote

A Friendly Introduction to Number Theory by Joseph H. Silverman. Although the proofs provided are fairly rigorous, the prose is very conversational, which makes for an easy read. Also, the material is presented so that even a student with a low to moderate level of mathematical maturity can follow the text conceptually and do many of the exercises, but there are plenty of exercises to stretch the more curious mathematician's mind.

As an undergrad I found it very useful and even years later it is one of my all-time favorite number theory references.

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Agree. If it had been around, I would not have felt the need to write a number theory book. – André Nicolas Sep 5 '12 at 1:43
up vote 4 down vote

One of my favorites is H. Davenport's ${\bf The\ Higher\ Arithmetic}$

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It is a very nice book. – André Nicolas Sep 5 '12 at 1:44
up vote 2 down vote

Elements of Number Theory by Andre Weil is the slickest,most concise yet best written introduction to number theory I've ever seen-it's withstood the test of time very well. For math students that have never learned number theory and want to learn it quickly and actively, this is still your best choice.

For more advanced readers with a good undergraduate background in classical analysis, Melvyn Nathason's Elementary Number Theory is outstanding and very underrated. It's very well written and probably the most comprehensive introductory textbook on the subject I know,ranging from the basics of the integers through analytic number theory and concluding with a short introduction to additive number theory, a terrific and very active current area of research the author has been very involved in.I heartily recommend it.

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up vote 2 down vote

I was shocked to see no one mentioned LeVeque's Fundamentals of Number Theory (Dover). He also authored Elementary Theory of Numbers with same publisher.

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up vote 2 down vote

Kato's "Fermat's Dream" is a jewel. (Full disclosure: actually I saw it mentioned either here or on mathoverflow, and I was looking for the post to thank the source.)

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up vote 0 down vote

For people interested in Computational aspects of Number Theory, A Computational Introduction to Number Theory and Algebra - Victor Shoup , is a good book. It is available online.

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