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Books on Number Theory for anyone who loves Mathematics?(Beginner to Advanced & just for someone who has a basic grasp of math)

30 Answers 30

125

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

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    @TheLastCipher The book is accessible without prior exposure to number theory, but I'd say a course in abstract algebra is necessary. Parts of the book also require concepts from analysis.2017-10-11
66

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

47

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

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    I think it is a pretty dry and dull book.2018-10-24
46

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

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    It is a good book but it requires a lot of time to get through this book as it has so many exercises. Is there a new edition2019-05-12
39

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.

28

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.2014-09-10
27

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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    @JoelCohen - it's not any more difficult than Serre's "A course in arithmetic," which has a lot of upvotes.2013-08-04
23

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.

17

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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    To be fair, the Haar Measure is used on page 3 of Basic Number Theory.2018-02-10
14

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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    I'm reading this book right now, it is a superb primer on the topic even for people without a strong math background.2017-02-01
12

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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    Is Cassels and Frohlich still a must? I had the impression that Neukirch, or Milne's notes http://www.jmilne.org/math/CourseNotes/cft.html were adequate substitutes, and perhaps more readable.2019-05-12
12

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.2012-12-12
11

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

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    @DonAntonio M. Ram Murty. See http://www.amazon.com/Problems-Algebraic-Number-Graduate-Mathematics/dp/03872218242014-10-25
11

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

11

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

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    It is a very nice book.2012-09-05
11

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.)

10

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

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    @user123 - There is only the one version.2015-06-16
8

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!2012-09-12
8

Manin and Panchishkin's Introduction to Modern Number Theory

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    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.2011-03-01
8

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.2012-09-05
8

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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    @Ihf Yes,my bad. : (2013-12-24
8

W.Sierpinski

Elementary Number Theory

From the master.

8

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.

7

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.

4

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.

3

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.2014-01-17
3

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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    It was good. Sagemath helped a lot with the programming, as it gave more time for compiling than other online free compilers. The number theory commands in Sage are powerful, though, so I was glad they wrote programs to investigate some of the early ideas first. But once we got to the totient function and the like, it really supported the students. Stein wrote a lot of the number theory routines for Sage, so the book was a perfect fit for that.2014-10-01
2

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!

1

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

1

elementary number theory by underwood dudley. This is cheap being a dover book (so no big financial investment) and has witty & imaginatively written proofs although it might be challenging to the complete beginner nevertheless there are hints for a lot of the exercises.