Books on Number Theory for anyone who loves Mathematics?
(Beginner to Advanced & just for someone who has a basic grasp of math)
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13$\begingroup$ @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. $\endgroup$– Charles StewartCommented Jul 21, 2010 at 13:59
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2$\begingroup$ @Prasoon: Those types of questions are typically also community wiki, for the same reason. $\endgroup$– Larry WangCommented Jul 21, 2010 at 14:36
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1$\begingroup$ This is absolutely absurd, why on earth are you linking to wikipedia? $\endgroup$– quantaCommented Apr 26, 2011 at 16:22
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7$\begingroup$ @quanta But what was the problem with the link to the Wikipedia article about number theory? $\endgroup$– Adrián BarqueroCommented Apr 26, 2011 at 16:42
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2$\begingroup$ Does this answer your question? Highschool level number theory book recommendations $\endgroup$– user53259Commented Jul 20, 2021 at 7:50
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32 Answers
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A Classical Introduction to Modern Number Theory by Ireland and Rosen hands down!
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10$\begingroup$ +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. $\endgroup$ Commented Sep 26, 2010 at 15:35
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9$\begingroup$ 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. $\endgroup$ Commented Sep 12, 2012 at 7:06
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7$\begingroup$ 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. $\endgroup$ Commented Jan 11, 2014 at 16:46
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2$\begingroup$ But why is it hands down the best? $\endgroup$ Commented Mar 7, 2015 at 11:18
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5$\begingroup$ @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. $\endgroup$– qoppaCommented Oct 11, 2017 at 22:12
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I would still stick with Hardy and Wright, even if it is quite old.
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21$\begingroup$ 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! $\endgroup$ Commented Apr 26, 2011 at 13:08
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8$\begingroup$ Holy shit an index? That's been something long missing from an Introduction to the Theory of Numbers. $\endgroup$ Commented May 6, 2014 at 19:51
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1$\begingroup$ I agree but -why? It is easy for everyone to find this book, why is it so good? $\endgroup$ Commented Mar 7, 2015 at 11:19
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1$\begingroup$ IMO, because it contains all the standard stuff. $\endgroup$– mauCommented Mar 9, 2015 at 20:08
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1$\begingroup$ The question I asked 4 years ago ("Why is an Introduction to the Theory of Numbers so good") looks so utterly silly now. - I do understand it now, because it brings you as close to G.H. Hardy and his contemporaries as possible. $\endgroup$ Commented May 12, 2019 at 13:47
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1$\begingroup$ The most recent edition, which is Niven, Zuckerman, and Montgomery, is even better than the earlier editions, which were very good. $\endgroup$ Commented Apr 26, 2011 at 13:06
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18$\begingroup$ 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. $\endgroup$ Commented Sep 5, 2012 at 2:30
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$\begingroup$ 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... $\endgroup$ Commented Dec 21, 2013 at 17:16
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$\begingroup$ 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 edition $\endgroup$– matqkksCommented Oct 24, 2018 at 14:28
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Serre's "A course in Arithmetic" is pretty phenomenal.
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8$\begingroup$ 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. $\endgroup$ Commented Aug 4, 2013 at 17:39
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3$\begingroup$ Why then is it pretty phenomenal? $\endgroup$ Commented Mar 7, 2015 at 11:20
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1$\begingroup$ I think it is a pretty dry and dull book. $\endgroup$– matqkksCommented Oct 24, 2018 at 14:27
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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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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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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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$\begingroup$ I totally agree. It's both accessible and up to date. But I'm not sure the book could be read as an introduction. $\endgroup$ Commented May 8, 2011 at 3:47
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$\begingroup$ @JoelCohen - it's not any more difficult than Serre's "A course in arithmetic," which has a lot of upvotes. $\endgroup$ Commented Aug 4, 2013 at 17:40
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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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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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3$\begingroup$ Agree. If it had been around, I would not have felt the need to write a number theory book. $\endgroup$ Commented Sep 5, 2012 at 1:43
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2$\begingroup$ I'm reading this book right now, it is a superb primer on the topic even for people without a strong math background. $\endgroup$ Commented Feb 1, 2017 at 9:10
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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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13$\begingroup$ 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). $\endgroup$– Matt ECommented Dec 31, 2010 at 4:31
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20$\begingroup$ Yes, Matt, me too, and it could explain why I have no memory at all of page 2. $\endgroup$ Commented Jan 5, 2011 at 20:03
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$\begingroup$ Who do you recommend the DA To? $\endgroup$ Commented Mar 7, 2015 at 11:22
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1$\begingroup$ To be fair, the Haar Measure is used on page 3 of Basic Number Theory. $\endgroup$– FraCommented Feb 10, 2018 at 4:00
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Elementary Number Theory - by David M. Burton if you want it somewhere halfway between fast and slow.
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3$\begingroup$ 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. $\endgroup$– ShreyaCommented Dec 12, 2012 at 11:53
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$\begingroup$ @Shreya ,...After reading this book then which book should i prefer to read so that i could take more deep understanding of this feild? $\endgroup$– user999691Commented Jun 24, 2022 at 19:24
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$\begingroup$ @JayendraandSankalp Depends on your background. If you are an undergraduate with decent Algebra background, then you can start reading something like Alaca & Willaims' Algebraic Number Theory. It has enough details for a beginner. Daniel Marcus's Number Fields is also really nice and is kinda self-contained. $\endgroup$– ShreyaCommented Jul 10, 2022 at 19:57
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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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$\begingroup$ Well, how about the Jurgen Neukirch? $\endgroup$– awllowerCommented Mar 1, 2011 at 11:38
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6$\begingroup$ 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. $\endgroup$ Commented Jun 29, 2011 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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$\begingroup$ @DonAntonio M. Ram Murty. See amazon.com/Problems-Algebraic-Number-Graduate-Mathematics/dp/… $\endgroup$ Commented Oct 25, 2014 at 3:53
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Basic Number Theory by Andre Weil. It's hard going and mind-blowing.
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11$\begingroup$ 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. $\endgroup$ Commented Apr 4, 2011 at 18:02
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3$\begingroup$ 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... $\endgroup$– awllowerCommented Apr 9, 2011 at 2:37
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16$\begingroup$ This is so far the funniest conversation I ever saw on MSE.(LoL) $\endgroup$ Commented May 18, 2012 at 11:34
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1$\begingroup$ 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. $\endgroup$ Commented Sep 12, 2012 at 7:09
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1$\begingroup$ 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. $\endgroup$ Commented Aug 4, 2013 at 17:42
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One of my favorites is H. Davenport's ${\bf The\ Higher\ Arithmetic}$
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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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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$\begingroup$ +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! $\endgroup$ Commented Sep 12, 2012 at 7:10
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$\begingroup$ This is probably the only answer here that OP would consider considering he's a programmer with a calculus background. $\endgroup$– OkoyosCommented May 10, 2021 at 17:30
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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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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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$\begingroup$ Do you mean Number Theory for Beginners by Weil with Rosenlicht? $\endgroup$– lhfCommented Dec 18, 2013 at 11:53
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$\begingroup$ broken link, as far as i can tell $\endgroup$ Commented Sep 26, 2010 at 17:31
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$\begingroup$ 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. $\endgroup$– awllowerCommented Mar 1, 2011 at 11:50
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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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2$\begingroup$ Not a really bad book, apart from the calligraphy, which is a truly terrible idea. $\endgroup$ Commented Sep 5, 2012 at 2:03
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$\begingroup$ Would you recommend that to use as a text for an introductory course for Math Majors on Elementary Number Theory? $\endgroup$ Commented May 12, 2019 at 13:51
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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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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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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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Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)
New answer to an old question:
An Illustrated Theory of Numbers, by Martin H. Weissman ($2017$)
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Stewart&Tall's "Algebraic Number Theory" is great.
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1$\begingroup$ 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. $\endgroup$ Commented Jan 17, 2014 at 12:17
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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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1$\begingroup$ How did the semester go? I'm thinking of using some of the Sage in a elementary number theory course teach in the spring. Currently, my plan was to use Jones and Jones Elementary Number Theory supplemented by some examples/exercises from Stein. Thoughts? $\endgroup$ Commented Sep 30, 2014 at 14:49
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$\begingroup$ 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. $\endgroup$ Commented Oct 1, 2014 at 15:18
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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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In my opinion, "the theory of numbers" by Neal H. Mccoy contains all number theory knowledge that a common person should have.