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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

29 Answers 29

up vote 69 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
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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
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From the first year undergraduate perspective, this book is worth a try but as @DanielTrebbien has noted, it is quite rigorous. Furthermore, it assumes that you know concepts such as rings to begin with. So, it can be difficult trying to understand everything the authors are trying to say. –  gekkostate Jan 11 at 16:46

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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Holy shit an index? That's been something long missing from an Introduction to the Theory of Numbers. –  Joshua Biderman May 6 at 19:51

Serre's "A course in Arithmetic" is pretty phenomenal.

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But I don't think Serre's book is at all suitable for "everyone who loves mathematics." It's definitely not a book that would be very helpful to the average undergrad. –  Daniel Miller Aug 4 '13 at 17:39

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
    
May I know what is the latest version of this book? I would love to read it. But I don't have a flair nor much of an interest in number theory... –  ireallydonknow Dec 21 '13 at 17:16

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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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
    
@JoelCohen - it's not any more difficult than Serre's "A course in arithmetic," which has a lot of upvotes. –  Daniel Miller Aug 4 '13 at 17:40

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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Absolutely, for beginners. –  vesszabo Sep 10 at 10:27

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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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
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+1 for the Disquisitiones Arithmeticae. –  awllower Mar 1 '11 at 11:41

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

Basic Number Theory by Andre Weil. It's hard going and mind-blowing.

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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
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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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Well,this response gets off on a technicality-I THOUGHT the question was asking for the best INTRODUCTION to the subject.Apparently not.Weil's book is NOT an elementary textbook anymore then Nathan Jacobson's BASIC ALGEBRA is a baby introduction to undergraduate algebra. –  Mathemagician1234 Sep 12 '12 at 7:09
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The first few chapters of Weil's book are fantastic, but I found the second half (on class field theory) very unappetizing. He totally (and intentionally) ignores the modern cohomological toolkit, which makes the statements (and proofs) of theorems nowhere near as clear as they could be. –  Daniel Miller Aug 4 '13 at 17:42

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

Problems in Algebraic Number Theory is written in a style I'd like to see in more textbooks

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Author?! ........ –  DonAntonio Jul 9 '13 at 18:18

For a highly motivated account of analytic number theory, I'd recommend Harold Davenport's Multiplicative Number Theory.

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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

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

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

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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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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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

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

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

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I am not sure if Stewart&Tall lives up to be "everyone who loves Mathematics should read". Though it is written as a first course in ANT and supposed to be an easy-read, it has some (at least three as I am aware of) logical gaps that may be difficult for beginners to fill in. And it has many annoying typos in Fraktur, especially in chapter 5, which distracts the reader. Of course they will not cause serious problems if it is used as a classroom text, where the instructor can provide corrections and relevant information. –  eltonjohn Jan 17 at 12:17

Number Theory For Beginners 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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Do you mean Number Theory for Beginners by Weil with Rosenlicht? –  lhf Dec 18 '13 at 11:53
    
@Ihf Yes,my bad. : ( –  Mathemagician1234 Dec 24 '13 at 3:54

Another interesting book: A Pathway Into Number Theory - Burn

[B.B] The book is composed entirely of exercises leading the reader through all the elementary theorems of number theory. Can be tedious (you get to verify, say, Fermat's little theorem for maybe $5$ different sets of numbers) but a good way to really work through the beginnings of the subject on one's own.

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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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Perhaps "best ever" is putting it a bit strong, but for me one of the best besides L E Dickson's books was "Elementary Number Theory" by B A Venkov, which does have an English translation.

One advantage of this book is that it covers an unusual and quite eclectic mix of topics, such as a chapter devoted to Liouville's methods on partitions, and some of these are hard to find in other texts.

The best benefit for me, paradoxically, was that the English translation I worked with was littered with misprints, in places a dozen or more per page. So after a while it became quite an enjoyable challenge to find them, and this meant having to study and consider the text more closely than one might have done otherwise!

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William Stein has shared his Elementary Number Theory online: http://wstein.org/ent/ It is accessible, lots of examples and has some nice computation integration using SAGE. I'll be using it this semester with secondary teachers, and will report back if things go particularly well or poorly with it.

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In my opinion, "the theory of numbers" by Neal H. Mccoy contains all number theory knowledge that a common person should have.

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