Books on Number Theory for Layman

Question

Books on Number Theory for anyone who loves Mathematics?

(Beginner to Advanced & just for someone who has a basic grasp of math)

@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
@Prasoon: Those types of questions are typically also community wiki, for the same reason. — Larry Wang, Jul 21 '10 at 14:36
This is absolutely absurd, why on earth are you linking to wikipedia? — quanta, Apr 26 '11 at 16:22
@quanta But what was the problem with the link to the Wikipedia article about number theory? — Adrián Barquero, Apr 26 '11 at 16:42
How can a question with 180 upvotes and 183 stars ever be closed? — nilo de roock, May 12 '19 at 13:41
nilo de roock It is closed by the mediocre and petty that can contribute nothing else but accumulate 'reputation' by trying to close posts and answers. — onepound, May 15 '21 at 10:28
Does this answer your question? [Highschool level number theory book recommendations](https://math.stackexchange.com/questions/725011/highschool-level-number-theory-book-recommendations) — NNOX Apps, Jul 20 '21 at 07:50

score 133 · Accepted Answer · edited May 31 '11 at 15:35

133

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

edited May 31 '11 at 15:35

claws

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answered Jul 22 '10 at 13:20

Moor Xu

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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 07:06
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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. – Jeel Shah Jan 11 '14 at 16:46
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But why is it hands down the best? – nilo de roock Mar 07 '15 at 11:18
Why don't you try reading it yourself? – Confuse Oct 03 '16 at 10:00
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Is this book accessible to a complete beginner? Considering I have not taken a course in abstract algebra or number theory before. – TheLast Cipher Sep 05 '17 at 17:51
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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. – qoppa Oct 11 '17 at 22:12
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it's not cheap (currently about 70 dollars)... – PatrickT Aug 16 '20 at 05:19
Does it come with a good amount of quality problems? – Choripán Con Pebre Sep 28 '21 at 02:18

score 72 · Answer 2 · answered Jul 21 '10 at 13:31

72

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

answered Jul 21 '10 at 13:31

mau

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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. – Stella Biderman May 06 '14 at 19:51
I agree but -why? It is easy for everyone to find this book, why is it so good? – nilo de roock Mar 07 '15 at 11:19
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IMO, because it contains all the standard stuff. – mau Mar 09 '15 at 20:08
@mau Could you have a look at my related question please. – advocateofnone Jan 20 '17 at 14:38
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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. – nilo de roock May 12 '19 at 13:47

Jason S · Answer 3 · 2015-03-20T15:29:01.107

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

edited Mar 20 '15 at 15:29

answered Jul 21 '10 at 14:25

Jason S

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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 05 '12 at 02:30
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
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 – matqkks Oct 24 '18 at 14:28

score 49 · Answer 4 · answered Jul 21 '10 at 14:35

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

answered Jul 21 '10 at 14:35

BBischof

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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 04 '13 at 17:39
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Why then is it pretty phenomenal? – nilo de roock Mar 07 '15 at 11:20
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I think it is a pretty dry and dull book. – matqkks Oct 24 '18 at 14:27

arsmath · Answer 5 · 2010-12-30T23:45:33.203

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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.

edited Dec 30 '10 at 23:45

answered Dec 30 '10 at 21:05

arsmath

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score 29 · Answer 6 · answered Jun 29 '11 at 03:07

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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.

answered Jun 29 '11 at 03:07

Tom Dickens

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Absolutely, for beginners. – vesszabo Sep 10 '14 at 10:27

score 26 · Answer 7 · answered Apr 26 '11 at 16:04

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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.

answered Apr 26 '11 at 16:04

Zev Chonoles

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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 08 '11 at 03:47
@JoelCohen - it's not any more difficult than Serre's "A course in arithmetic," which has a lot of upvotes. – Daniel Miller Aug 04 '13 at 17:40

score 23 · Answer 8 · answered Jul 21 '10 at 14:03

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.

answered Jul 21 '10 at 14:03

user115

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score 18 · Answer 9 · edited Apr 22 '14 at 10:33

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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.

edited Apr 22 '14 at 10:33

Assad Ebrahim

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answered Dec 30 '10 at 23:08

roy smith

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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 04:31
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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 05 '11 at 20:03
Who do you recommend the DA To? – nilo de roock Mar 07 '15 at 11:22
To be fair, the Haar Measure is used on page 3 of Basic Number Theory. – Fra Feb 10 '18 at 04:00

score 17 · Answer 10 · answered Sep 05 '12 at 00:16

17

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.

answered Sep 05 '12 at 00:16

pocketsand

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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 05 '12 at 01:43
I'm reading this book right now, it is a superb primer on the topic even for people without a strong math background. – Fabrizio Calderan Feb 01 '17 at 09:10

score 15 · Answer 11 · answered Jan 05 '11 at 20:25

15

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

link

answered Jan 05 '11 at 20:25

Myself

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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. – Shreya Dec 12 '12 at 11:53

score 13 · Answer 12 · answered Sep 26 '10 at 15:31

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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.

answered Sep 26 '10 at 15:31

Andrea Mori

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Well, how about the Jurgen Neukirch? – awllower Mar 01 '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 http://www.jmilne.org/math/CourseNotes/cft.html were adequate substitutes, and perhaps more readable. – David E Speyer Jun 29 '11 at 03:11

score 11 · Answer 13 · answered Dec 30 '10 at 23:07

11

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

answered Dec 30 '10 at 23:07

Barry Smith

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It is already mentioned. – awllower Mar 01 '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 04 '11 at 18:02
Excuse moi? What are you referring to? – awllower Apr 05 '11 at 06:04
I was the first one to mention this book. The only other person to mention it (at least in a comment or answer that is still visible) is Roy Smith. And he only mentioned it in reference to my comment right here, as clearly indicated in his comment. I was the first one to mention Weil's book, so you are wrong, it had not been mentioned at the time of my post; and even if it had been, why would you come along 3 months later and point it out? – Barry Smith Apr 08 '11 at 19:31
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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 09 '11 at 02:37
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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 07:09
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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 04 '13 at 17:42
What sentence, paragraph or theorem did you find "mind-blowing"? – nilo de roock May 12 '19 at 13:49

score 11 · Answer 14 · answered Sep 05 '12 at 00:31

11

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

answered Sep 05 '12 at 00:31

i. m. soloveichik

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It is a very nice book. – André Nicolas Sep 05 '12 at 01:44

score 11 · Answer 15 · answered Feb 04 '13 at 21:12

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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.)

answered Feb 04 '13 at 21:12

score 11 · Answer 16 · answered Jul 28 '10 at 17:53

11

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

answered Jul 28 '10 at 17:53

Simon Morris

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@DonAntonio M. Ram Murty. See http://www.amazon.com/Problems-Algebraic-Number-Graduate-Mathematics/dp/0387221824 – Gamma Function Oct 25 '14 at 03:53

score 10 · Answer 17 · answered Jul 31 '10 at 20:18

10

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

answered Jul 31 '10 at 20:18

Lucky

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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 07:10
This is probably the only answer here that OP would consider considering he's a programmer with a calculus background. – Okoyos May 10 '21 at 17:30

score 10 · Answer 18 · answered Jun 29 '11 at 00:11

10

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

answered Jun 29 '11 at 00:11

John M

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Is there only a graduate version or an undergraduate does also exist? @JohnM – abcdef Jun 13 '15 at 14:08
@user123 - There is only the one version. – John M Jun 16 '15 at 17:44

score 9 · Answer 19 · edited Jun 12 '20 at 10:38

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.

Do you mean *Number Theory for Beginners* by Weil with Rosenlicht? — lhf, Dec 18 '13 at 11:53

score 8 · Answer 20 · answered Apr 26 '11 at 13:10

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.

answered Apr 26 '11 at 13:10

Gerry Myerson

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Not a *really* bad book, apart from the calligraphy, which is a truly terrible idea. – André Nicolas Sep 05 '12 at 02:03

score 8 · Answer 21 · answered Jun 12 '13 at 13:21

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W.Sierpinski

Elementary Number Theory

From the master.

answered Jun 12 '13 at 13:21

Torsten Hĕrculĕ Cärlemän

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Would you recommend that to use as a text for an introductory course for Math Majors on Elementary Number Theory? – nilo de roock May 12 '19 at 13:51

score 8 · Answer 22 · answered Sep 26 '10 at 12:24

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Manin and Panchishkin's Introduction to Modern Number Theory

answered Sep 26 '10 at 12:24

Who

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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 01 '11 at 11:50

score 8 · Answer 23 · answered Dec 12 '13 at 14:30

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.

score 7 · Answer 24 · answered Sep 12 '12 at 07:42

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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.

answered Sep 12 '12 at 07:42

Sniper Clown

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score 5 · Answer 25 · answered Sep 12 '12 at 06:45

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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.

answered Sep 12 '12 at 06:45

rrampage

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score 3 · Answer 26 · answered Jul 21 '10 at 16:49

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

answered Jul 21 '10 at 16:49

Noah Snyder

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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 '14 at 12:17

score 3 · Answer 27 · answered May 06 '14 at 19:45

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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.

answered May 06 '14 at 19:45

John Golden

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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? – James S. Cook Sep 30 '14 at 14:49
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. – John Golden Oct 01 '14 at 15:18

score 2 · Answer 28 · answered Jul 26 '21 at 02:30

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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 ($2017$)

answered Jul 26 '21 at 02:30

J. W. Tanner

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score 2 · Answer 29 · edited Dec 18 '13 at 13:32

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!

score 1 · Answer 30 · edited May 10 '21 at 17:33

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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.

edited May 10 '21 at 17:33

Okoyos

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answered Oct 16 '17 at 18:50

onepound

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score 1 · Answer 31 · answered Jul 26 '21 at 04:20

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For someone who wants to see a lot of neat Number Theory facts and isn't too interested in proofs, Albert Beiler, Recreations in the Theory of Numbers, is perfect.

answered Jul 26 '21 at 04:20

Gerry Myerson

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score 1 · Answer 32 · answered May 28 '14 at 18:01

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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.

answered May 28 '14 at 18:01

fernanfio

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Books on Number Theory for Layman

32 Answers32

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