Reddit mentions: The best number theory books

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)

By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.

So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.

Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.

Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.

 




Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.

So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?

Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis

Number Theory. This starts out kinda easy but blows up in your face, I've been looking into L-functions a lot recently and they just make me feel so inadequate as a mathematician; contains exceptional amounts of Representation Theory, Functional Analysis, Real/Complex Analysis, Algebraic Geometry, Algebraic Number Theory, p-adic Analysis, Elliptic Curves, Automorphic Forms, Class Field Theory, Riemann-Roch Theory, Etale-Cohomology... crazy!

Anyhow, a lot of Algebraic Number Theory books spend a lot of time in the beginning defining Number Fields and extensions, going into ramification, splitting of primes etc. I think people need to learn a bit more about solving Diophantine Equations in Z and Q before looking at general number fields. So, if D(x_1,...,x_n)=0 is a Diophantine Equation, then we can look at D_p^k (x_1,...,x_n)=0, where everything reduced mod p^k . Clearly if D has a solution in Z, then each D_p^k will have a solution. Now, there are a lot of p^k 's, but luckily we can reduce the number of equations we have to check by looking at the p-adic fields. It turns out that D_p^k has a solution iff the D^p has a solution (where D^p is D considered as an equation in the p-adic field). So, this suggests that we try and figure out if an equation is solvable in all the p-adics, then is it solvable in Q?

It turns out that this is true, when D is quadratic (each term has total degree at most 2) with the added stipulation that D also have a solution in R, this is called the Hasse Principle. Fermat's Theorem gives a counter example for the case when the degree is greater than 2, since x^n +y^n -z^n =0 is solvable in all the p-adics and R. The appearance of R in this theorem is the first suggestion that something about R behaves like a prime, and it is generally treated as a "prime at infinity".

The thing I really want to call your attention to is the fact that we are looking at each prime individually to see if there is a solution on the whole. Looking at the p-adics means we are looking "locally", and looking at Q means we are looking "globally". This is justified in the sense that we can add a topology to the set of prime numbers that makes each prime a local entity and the whole of Z is the global thing. This kind of Local-to-Global principal is viral throughout mathematics, Category Theory formalized treatment like this with the concept of a Sheaf. I just think it is very cool that this extremely geometric principal, that lays the foundation for Differential Geometry (things are nice locally, so we see what happens there and figure out what we can extrapolate globally), has been very fruitful in the study of a field that (initially) seems as far as you can get from DG.

L-Functions, Zeta-Functions etc are studied by trying to express them locally in terms of primes to figure out their global properties. For example, the Riemann-Zeta Function is defined by a sum over all the powers of integers, but Euler proved that it can, instead be written as an infinite product of functions relating to the primes, giving a local interpretation to a function defined globally. Using this new Euler Decomposition, we can prove important analytic properties about the Zeta-Function and it's nifty functional equation (that secretly contains info about the "prime at infinity").

For more info on Number Theory, especially at the advanced undergrad level, I recommend A Classical Introduction to Modern Number Theory, great fucking book.

I would say that it would depend on the problem. If you cannot solve the first ten, I would be worried, as they can all be solved by simple brute force methods. I have a degree in Astrophysics, and some of the 300 and 400 problems are giving me pause, so if you are stuck there you are in good company.

There are elegant solutions to each problem, if you want to delve into them, but the first handful, the first ten especially, can be simply solved.

Once you get beyond the first ten or so, the mathematical difficulty ratchets up. There are exceptions to that rule of course, but by and large, it holds.

If you are interested in Number Theory, the best place to start is a number theory course at a local university. Mathematics, especially number theory, is difficult to learn by yourself, and a good instructor can expound, not only on the math, but also on the history of this fascinating subject.

Gauss, quite arguably the finest mathematician to ever live loved number theory; of it, he once said:

> Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

Although my personal favorite quote of his on the subject is:

> The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it.

If you are interested in purchasing some books about number theory, here are a handful of recommendations:


Number Theory (Dover Books on Mathematics) by George E. Andrews


Number Theory: A Lively Introduction with Proofs, Applications, and Stories by James Pommersheim, Tim Marks, Erica Flapan


An Introduction to the Theory of Numbers by G. H. Hardy, Edward M. Wright, Andrew Wiles, Roger Heath-Brown, Joseph Silverman


Elementary Number Theory (Springer Undergraduate Mathematics Series) by Gareth A. Jones , Josephine M. Jones

and it's companion


A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (v. 84) by Kenneth Ireland, Michael Rosen

and a fun historical book:


Number Theory and Its History (Dover Books on Mathematics) Paperback by Oystein Ore

I would also recommend some books on

Markov Chains

Algebra

Prime number theory

The history of mathematics

and of course, Wikipedia has a good portal to number theory.

Oi. Disclaimer: I haven't bought a book in the field in a while, so there might be some new greats that I'm not familiar with. Also, I'm old and have no memory, so I may very well have forgotten some greats. But here is what I can recommend.

I got my start with Koblitz's Course in Number Theory and Cryptography and Schneier's Applied Cryptography. Schneier's is a bit basic, outdated, and erroneous in spots, and the guy is annoying as fuck, but it's still a pretty darned good intro to the field.

If you're strong at math (and computation and complexity theory) then Oded Goldreich's Foundations of Cryptography Volume 1 and Volume 2 are outstanding. If you're not so strong in those areas, you may want to come up to speed with the help of Sipser and Moret first.

Also, if you need to shore up your number theory and algebra, Victor Shoup is the man.

At this point, you ought to have a pretty good base for building on by reading research papers.

One other note, two books that I've not looked at but are written by people I really respect Introduction to Modern Cryptography by Katz and Lindell and Computational Complexity: A Modern Approach by Arora and Barak.

Hope that helps.

As I see it there are four kinds of books that fall into the sub $30 zone:

• Dover books which are generally pretty good and cover a wide range of topics
  
  
• Free textbooks and course notes - two examples I can think of are Hatcher's Algebraic Topology (somewhat advanced material but doable if you know basic point-set topology and group theory) and Wilf's generatingfunctionology
  
  
• Really short books—I don't a good example of this, maybe Stanley's book on catalan numbers?
  
  
• Used books—one that might interest you is Automatic Sequences by Allouche and Shallit
  
  You can get a lot of great books if you are willing to spend a bit more however. For example, Hardy and Wright is an excellent book (and if you think about it: is a 600 page book for $60 really more expensive than a 300 page one for 30?). Richard Stanley's books on combinatorics: Enumerative Combinatorics Vol. I and Algebraic Combinatorics are also excellent choices. For algebra, Commutative Algebra by Eisenbud is great. If computer science interests you you can find commutative algebra books with an emphasis on Gröbner bases or on algorithmic number theory.
  
  So that's a lot of suggestions, but two of them are free so you can't go wrong with those.

Depends what kind of math you're interested in. If you're looking for an introduction to higher (college) math, then How to Prove It is probably your best bet. It generally goes over how proofs work, different ways of proving stuff, and then some.

If you already know about proofs (i.e. you are comfortable with at least direct proofs, induction, and contradiction) then the world is kind of your oyster. Almost anything you pick up is at least accessible. I don't really know what to recommend in this case since it's highly dependent on what you like.

If you don't really know the basics about proofs and don't care enough to yet, then anything by Dover is around your speed. My favorites are Excursions in Number Theory and Excursions in Geometry. Those two books use pretty simple high school math to give a relatively broad look at each of those fields (both are very interesting, but the number theory one is much easier to understand).

If you're looking for high school math, then /u/ben1996123 is probably right that /r/learnmath is best for that.

If you want more specific suggestions, tell me what you have enjoyed learning about the most and I'd be happy to oblige.

Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.

I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.

I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.

In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.

There is Elementary number theory by William Stein, and A Computational Introduction to Number Theory and Algebra. The latter is better if you are also interested in some of the computation They are both available for free online (legally). I think you would prefer Stein's book but skim through both and see which one you like more.

For something more in depth, I looked at some of the books in this list at mathoverflow. Hardy & Wright , and Niven & Zuckerman's books seem best suited to you (from what I looked at, but go through that list yourself). Many of the other books require some background in abstract algebra.

I haven't read either but just looking through their table of contents I would go with Niven and Zuckerman's book. It seems to go into the more useful things more quickly, and it's not as densely packed with information you probably won't be interested in right now.

TLDR: Start here, or here.

For what it's worth, number theory is a fascinating field. I don't think you'll be disappointed going into it. Good luck!

It's a bit complicated and I can't claim to be an expert but here's what I'm familiar with.

Group theory is the idea that certain systems can be represented as a closed group governed by some operator. If you apply the operator to some member of the group, you will always get another member of the group. G(Z, +), a group of integers governed by the + operator, is a group because you can add 1 (or subtract 1 -- the reverse of the operator) continuously to get every other integer.

Field theory governs fields, which are basically sets of numbers in which addition, subtraction, multiplication, and division are defined (so all of your favorite number systems -- real numbers, natural numbers, the whole family -- they're fields). Galois theory is dedicated to re-interpreting fields as groups to better understand them. The single operator of groups typically gives a better way of describing the relationship of numbers within a set than field theory does.

I wish I had a concise example to give but I really can't think of one. If you want to read more about it though I'd recommend Fearless Symmetry.

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

At some point these "Pop" reading books get wholly unsatisfying and you need textbooks, but I think that's a story for a different semester. Theres a good set of books written by Avner Ash and Robert Gross (Boston College) that anyone with calculus 1 can easily get into:
Elliptic Tales:
https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199
Fearless Symmetry:
https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710/ref=pd_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=JG1NQ2F2XS0WJJ5PBKVV

Well worth the read, entertaining, and great introductions to their respective subjects!

Well, Hardy & Wright is the classic book for elementary stuff. It has almost everything there is to know. There is also a nice book by Melvyn Nathanson called Elementary Methods in Number Theory which I really like and would probably be my first recommendation. Beyond that, you need to decide which flavour you like. Algebraic and analytic are the big branches.

For algebraic number theory you'll need a solid grounding in commutative algebra and Galois theory - say at the level of Dummit and Foote. Lang's book is pretty classic, but maybe a tough first read. I might try Number Fields by Marcus.

For analytic number theory, I think Davenport is the best option, although Montgomery and Vaughan is also popular.

Finally, Serre (who is often deemed the best math author ever) has the classic Course in Arithmetic which contains a bit of everything.

Excursions in Number Theory by Ogilvy and Anderson manages to touch on a lot of the most beautiful and interesting results of elementary number theory with almost no prerequisite knowledge. Number theory is a really concrete and easily visualized field of study and doesn't rely on a lot of abstraction like some other fields do. The proofs are also in general pretty simple but illuminating. It was my first math book. Definitely 10/10 recommend.

Edit: I think there are a few others in the Excursions series too, which makes reading about several fields simultaneously pretty easy, what with consistent notation/style and all.

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

Agreed.
Lay's was the textbook for my proofs course and I was rather pleased with it. Also because it gently introduces you to analysis, you will be more prepared than other students when you take analysis.

I would also possibly suggest an intro to number theory book. Problems in number theory are typically very easy to comprehend (and I think rather fun), giving you a chance to adopt a more proof-based thought pattern without also being burdened with too many theorems and definitions. Many intro to number theory books (like this one)
have exercises that encourage you to find/prove patterns and theorems yourself, exercises which are really nice to think about and digest in your leisure time rather than have to think about with a homework deadline in mind.

For classes like number theory and abstract algebra, I would suggest just picking up a book and attempting to read it. It will be hard, but the main prerequisite for courses like this is some mathematical maturity. That only comes with practice.

Realistically there is probably no preparation that you could have which would prepare you in such a way that a book on advanced mathematics would be super easy.

I like this number theory book
http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1348165257&sr=8-1&keywords=number+theory

I like this abstract algebra book
http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_2?s=books&ie=UTF8&qid=1348165294&sr=1-2&keywords=abstract+algebra

There's so much I want to say, but I have to go to bed. For now let me leave you with these:

This is a great book. It's probably the most accessible book on this subject that you'll find.

For a quicker read that still gives some motivation for these things, there's this paper written by a (then) undergraduate.

Finally, while I don't find the visuals extremely enlightening, this has pretty much everything you could ever want to know on that subject.

Hi, a similar question was asked a couple days ago. I recommend reading GOD_Over_Djinn's excellent explanation here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgyd5 or my own comment here: http://www.reddit.com/r/math/comments/1h2i9v/playing_around_with_an_idea_related_to_prime/caqgh42. The best way to learn about p-adic numbers is of course to read a book about them instead of just looking at wikipedia or reading what random people on the internet have to say. I cannot recommend enough Robert's "A Course in p-adic Analysis" if you have a basic knowledge of topology and analysis http://www.amazon.com/Course-p-adic-Analysis-Graduate-Mathematics/dp/0387986693. If you're more interested in p-adic zeta functions etc. look at Koblitz's "p-adic Numbers, p-adic Analysis and Zeta Functions" http://www.amazon.com/Numbers-Analysis-Zeta-Functions-Graduate-Mathematics/dp/1461270146/ref=sr_1_8?s=books&ie=UTF8&qid=1372366949&sr=1-8&keywords=p-adic+analysis. Although I haven't personally read it this book here also seems to be a more elementary introduction: http://www.amazon.com/p-adic-Numbers-Fernando-Quadros-Gouvea/dp/3540629114/ref=sr_1_3?s=books&ie=UTF8&qid=1372367005&sr=1-3&keywords=p-adic+analysis. The first 2 I know you can find pdfs of online. I don't know about the third. Alternatively, p-adic numbers are covered in a less technical sense in Bartel's notes on number theory here: http://homepages.warwick.ac.uk/~maslan/numthry.php. I haven't looked at them yet but I can say that his notes on representation theory are very good.

Any other construction I can think of aside from what I linked requires group theory or topology so its kinda hard unless you have a background in these subjects.

Edit: Having skimmed through Bartel's notes: they are an excellent introduction to p-adic numbers and he thoroughly covers them and their applications. I do recommend it.

This one's well-known and highly regarded as a good source.

I'm also going to start learning number theory because it's a pretty fun subject. So far, Hardy's been pretty good (I've only read excerpts of the 1st chapter though).

As for your background, you would only need to know basic facts about numbers (divisibility/primes etc) when starting Hardy so you should be fine I think.

Mathematics papers aren't really a good place to get an introduction to branches of mathematics as they tend to cater to the advanced reader. The most accessible you might find would be Mathematics Magazine or similar.

You would fare far better with an undergraduate level textbook. Springer publish a lot of pretty good undergraduate level texts so you might find something like this or this helpful (although I have not personally read either of those specifically so I cannot speak to their quality, but I find Springer books are usually good).

You might get better advice asking in the main sub (/r/math) as people like to give reading there.

EDIT: or maybe something like this would be more suited to what you're looking for?

How much is it worth to you? The following book is expensive, but its a great, well-written, undergrad-level survey:

http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469

You can look at your local book store for something cheaper, there's probably a Dover paperback on number theory for less than $20, but that's a crap-shoot. Good Luck!

What area of mathematics you love? Geometry? Number theory?

In Number theory, I recommend J.H. Silverman A friendly introduction to number theory, it's a good book that let me jump to solid mathematics from mathematics history book.

Alex Bello's Here's Looking at Euclid (something different in the UK, I think it's something about Mathmagic Land) is pretty good. Also, some of the videos from TED about math education are very good. Another video where it takes you through the calculations needed to find the number of pennies in a pyramid.

I'll try to find the links:

http://www.amazon.com/Heres-Looking-Euclid-Counting-Awe-Inspiring/dp/1416588280#

http://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en

Also, Dan Meyer has a website that's pretty good (http://blog.mrmeyer.com) and you can find good videos on Youtube about real world math problems.

Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).

Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.

Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.

Last term, on a whim I took an elective math course called "Introduction to Number Theory and Cryptography". The crypto didn't come til the end, but by then, with such an extensive background in number theory, it was easy, intuitive, and I understood.
We used Elementary Number Theory which I just realized is written by Kenneth Rosen (Discrete Math and it's applications). The book was great, and by the later chapters, Crypto wasn't just a list of algorithms, and oh good, they work- I actually understood it, I grokked it.

This term, I have a mandatory course in Crypto run by the CS department. It's just "here is the algorithm" over and over, with only a bit of background math proving things. We're using Cryptography: Theory and Practice, which has a lot of algorithms, and descriptions, but doesn't necessarily provide the rigorous math proofs I prefer.

A good first course book is Niven and covers a lot of ground at a superficial level. You will probably have a feeling for what you like after that.

If you want to pursue algebraic number theory after, you will need to study Galois theory. If you want to pursue analytic number theory, you will need some complex analysis. You will basically need to know the major results and have an intuition for them before starting number theory proper.

A fantastic second book is "Ireland and Rosen". I highly recommend this and it covers both aspects of number theory and then some.

Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.

As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.

Sounds like what you want is elementary number theory, which fortunately is something you can get in to without any real prerequisites.

There are some decent textbook recommendations here that you could try out (although as a warning, not all of those are for elementary number theory, check the descriptions). Perhaps Silverman's book would be a good one to try out, though there are certainly lots of other options if you'd rather try a different one. [Edit:Looks like velcrorex suggested the exact same book. I've definitely heard good things about it.]

> This seems very confusing to me, as it is defining p-adic expansion of numbers in terms of p-adic numbers ...

This is just a hand-wavy, intuitive explanation of what p-adic numbers look like. The fact is that once you formalize everything about the [p-adic valuation](http://en.wikipedia.org/wiki/P-adic_valuation) and the p-adic numbers, it turns out that every p-adic number has the series expansion that you mentioned.

> For instance, why, in the p-adic world, are positive powers of p small, and negative powers large? It seems like a prime number to a large power would be large, no?

When dealing with p-adic numbers, you have to forget all your intuition about the usual notions of absolute values and ordering of the real numbers, since they don't apply. Everything in the p-adic world is based on the p-adic valuations, which give their own topologies and notions of size. The p-adic topologies are very different from the topology on R. For example, any point within an open ball in the p-adic numbers can be considered that ball's center. Quirky things like this make it initially hard to grasp the concepts of p-adic numbers and their associated arithmetic, but once you practice working with them enough, they start to make sense.

> How does the limit of the sequence that they're talking about equal 1/3?

This again has to do with the fact that convergence in the p-adic topology is different from convergence in the usual Euclidean topology.

Some good resources for learning more about p-adic numbers are the following:

1. Gouvêa, Fernando Quadros, p-adic Numbers: An Introduction (Amazon, SpringerLink)
   
2. Koblitz, Neal, p-adic Numbers, p-adic Analysis, and Zeta-Functions (Amazon, SpringerLink)
   
3. Robert, Alain M, A Course in p-adic Analysis (Amazon, SpringerLink)
   
4. Serre, Jean-Pierre, A Course in Arithmetic (Amazon, SpringerLink)
   
   For me personally, learning general valuation theory was very useful for understanding p-adic numbers.

They may not be the best books for complete self-learning, but I have a whole bookshelf of the small introductory topic books published by Dover- books like An Introduction to Graph Theory, Number Theory, An Introduction to Information Theory, etc. The book are very cheap, usually $4-$14. The books are written in various ways, for instance the Number Theory book is highly proof and problem based if I remember correctly... whereas the Information Theory book is more of a straightforward natural-language summary of work by Claude Shannon et al. I still find them all great value and great to blast through in a weekend to brush up to a new topic. I'd pair each one with a real learning text with problem sets etc, and read the Dover book first quickly which introduces the reader to any unfamiliar terminology that may be needed before jumping into other step by step learning texts.

Liebeck's Concise Introduction to Pure Mathematics is a great text for introducing students to the basic tools required in abstract algebra, number theory and analysis, but doesn't go into great depth.

It's kind of a standard text but for abstract algebra I think Dummit and Foote is remarkably clear.

Ireland and Rosen's Classical Introduction to Modern Number Theory is a classic, but maybe more intermediate.

Elementary Number Theory by Jones is very good.

I think an introductory book on number theory would be useful, probably more immediately useful than a book on abstract algebra. Algebra is something you'd want to look into eventually, but you don't need all of the associated machinery to learn about Z/nZ, and algebra may be more abstract that you want, as an introduction.

I had Rosen, which was okay, but not great. Definitely not worth the price. Still, it may be useful to see the typical ToC of an intro number theory book so you can try to find a better one.

I like reading math books for fun, especially cheap Dover books. Excursions in Number Theory by Ogilvy & Anderson (lots of cool little stuff). Introductory Graph Theory by Chartrand (a lot of real-world programming boils down to graph theory). An Introduction to Algebraic Structures by Landin (abstract algebra). In Code: A Mathematical Journey by Flannery (modular arithmetic, factoring, and cryptography). In Code or Excursions would probably help with Project Euler 3 and several others.

Introduction to Analytic Number Theory by Apostol is a great introduction to analytic number theory. This would be a great way to tie together the number theory, combinatorics, and calculus that you've seen so far.

The best book punnily titled after a great thinker since Here's Looking at Euclid!

Yep the above book is great. Other good books that are more towards the math side:

guide to elliptic curve cryptography by menenzes

a course in number theory and cryptography

Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.

I have the Jones and Jones book. Since it wasn't my introduction to number theory, I can't say if it's a good introduction. Personally, I would go with another book, like the many already mentioned in other replies.


If you are looking for a nice cheap (free!) book, check out Stein's book Elementary Number Theory

I learned out of Niven et al, and thus it is my goto reference. But, it seems prohibitively expensive now: An Introduction to the Theory of Numbers

Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley

As in financial scamming? Yeah, that's low-effort in my eyes. Sorry OP, but you're going to need a better idea. If you can find a copy of [this] (http://www.amazon.com/Mathematical-Ideas-Really-Need-Series/dp/1847240089) book, take a look. You might find something interesting.

Maybe you'd like A Friendly Introduction to Number Theory by Silverman.

Also I highly recommend Gilbert Strang's textbooks Introduction to Applied Math and also Linear Algebra and its Applications.

There are a lot of canonical texts on both subjects. One of the classics on ant is Neukirch.

You should definitely find a book which has a good coverage of transcendence degree and transcendence basis. It's a simple concept but really enlightening.

I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.

The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.

Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.

If you have sufficient background in number theory, Koblitz's book is excellent. It's accessible to a strong undergraduate.

Hardy and Wright, An Introduction to the Theory of Numbers. Awesome book.

http://www.amazon.com/An-Introduction-Theory-Numbers-Hardy/dp/0199219869

I see from your edit that you found the Chevalley-Warning theorem. But if you are still interested in getting more detailed information about the solutions, it seems that there are some interesting regularities. If your quadratic form is nondegenerate (in the sense defined below) then it appears that there are always exactly p^2 solutions, including the zero solution.

First of all, the book of Jean-Pierre Serre, A Course of Arithmetic treats these matters in considerable detail, including a proof of the Chevalley-Warning theorem (page 16). But he also goes on to prove a classical result (page 34, due to Gauss maybe?), which shows that by a linear change of variables any quadratic form is equivalent to one with no off-diagonal terms. So you can reduce

ax^2 + by^2 + cz^2 + 2(exy + fxz + gyz) to a'x'^2 + b'y'^2 + c'z'^2

It is possible that the quadratic form is degenerate in the sense that one or more of a',b',c' turn out to be zero. However if none of the a', b' and c' are zero mod p, then it appears from my empirical tests that there are always p^2 solutions. You can experiment with this yourself if you have access to Mathematica. For example,

a = 1;
b = 1;
c = 1;
p = 7;
Clear[x, y, z]
solns = Solve[a x^2 + b y^2 + c z^2 == 0, {x, y, z}, Modulus -> p];
Length[solns]

The number of solutions, including the zero solution is 49 as claimed. I've tried a number of different prime moduli and various non-zero values for a, b, and c, and always gotten p^2

I haven't thought too much about how to prove all of this, but I thought you might still be interested.

p.s. What sort of imbecile would downvote a real mathematics post like this?

Rosen's Modern Number Theory (hands down) for number theory and Halmos' Finite-Dimensional Vector Spaces for linear algebra. There are other texts available for linear algebra but Halmos is a pretty solid option.

I would strongly recommend Apostle's intro to analytic number theory. It doesn't assume any prior knowledge of number theory, but does move a bit fast.

Some folks would probably suggest you get a book solely dedicated to elementary number theory, but the danger is that you can get bogged down in in the details and never get to the good stuff.

My university used George Andrews's book, which is Dover and really cheap. It was a pretty good book.

If you're not afraid of math there are some cheap introductory textbooks on topics that might be accessible:
For abstract algebra: http://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&qid=1459224709&sr=8-1&keywords=book+of+abstract+algebra+edition+2nd

For Number Theory: http://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=sr_1_1?ie=UTF8&qid=1459224741&sr=8-1&keywords=number+theory

These books have complimentary material and are accessible introductions to abstract proof based mathematics. The algebra book has all the material you need to understand why quintic equations can't be solved in general with a "quintic" formula the way quadratic equations are all solved with the quadratic formula.

The number theory book proves many classic results without hard algebra, like which numbers are the sum of two squares, etc, and has some of the identities ramanujan discovered.

For an introduction to analytic number theory, a hybrid pop/historical/textbook is : http://www.amazon.com/Gamma-Exploring-Constant-Princeton-Science/dp/0691141339/ref=sr_1_1?ie=UTF8&qid=1459225065&sr=8-1&keywords=havil+gamma

This book guides you through some deep territory in number theory and has many proofs accessible to people who remember calculus 2.

I enjoyed reading this as an adult, wish I would have read it in my teens or younger.

A Pathway into Number Theory by Burns might appeal to you. You might want to put extra effort into digging up a book that approaches elementary number theory from a combinatorial point of view, which is more in line with the stuff you're doing now.

EDIT: This seems perfect for you: https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/

While you are probably not going to understand the gist of papers on number theory by reading an introductory book, the best book, to my mind, is Ireland and Rosen.

There’s a great Radiolab about this by Alex Bellos: http://www.radiolab.org/story/love-numbers/

Alex has written a couple books about math and numbers: https://www.eamazon.com/Heres-Looking-Euclid-Counting-Awe-Inspiring/dp/1416588280/ref=la_B00K042IJC_1_2?s=books&ie=UTF8&qid=1522425983&sr=1-2

https://www.amazon.com/Grapes-Math-Reflects-Numbers-Reflect/dp/1451640110/ref=la_B00K042IJC_1_5?s=books&ie=UTF8&qid=1522425983&sr=1-5

This book has a great non-technical chapter about this.

Assuming you liked Apostol's intro, he wrote another text which is almost like a second volume: https://www.amazon.com/dp/0387971270/?tag=stackoverfl08-20

I like this book http://www.amazon.com/p-adic-Numbers-An-Introduction-Universitext/dp/3540629114

You should look at all the books on p-adic numbers in your library and find one you like.

I'm no expert either, but I highly recommend Apostol's Modular Functions and Dirichlet Series in Number Theory. It can be read after a first complex analysis course and gives a thorough background to the j-function in the first four chapters. There's nothing on moonshine, but the connections between modular functions and number theory begin here.

Well without knowing your background it's pretty hard to give you a recommendation... Ireland and Rosen is a classic. Note that this is a different book (and author) from /u/FunkMetalBass's comment.

I used this book when I was in high school:

Number Theory

Costs $8, explains things beautifully

That book is probably what you want. It looks like it focuses more on math and how it applies to cryptography rather than on crypto algorithms and how they work, pros/cons, etc. It was also used in this math class at Berkeley (lots of extra reading material on that page too).

Again, I think the book you found is what you want. But here are some other options if you want some:

• You may like A Course in Number Theory and Cryptography. A bit old, but still relevant.
  
  
• Intro to Modern Cryptography is what is used at my university and various others for an undergrad crypto course. Pretty rigorous.
  
  
• I doubt you'll want this, but Applied Cryptography is the seminal text on the subject. Most anything you could want to know about crypto is going to be in here (though it could use an update).
  
  

https://www.amazon.com/Number-Theory-Dover-Books-Mathematics/dp/0486682528/ref=pd_lpo_sbs_14_t_0/130-7956659-8948968?_encoding=UTF8&psc=1&refRID=7N6E11C8GDASPV8X3BAN

This was my first arithmetic book. I found it quite good and covered a wide range of topics.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

I'd recommend the following:

(1) Either of the Ash and Gross books Fearless Symmetry of Elliptic Tales

(2) Anything by Paul Nahin for instance Dr. Euler's Fabulous Formula

(3) Get yourself a Rubik's cube and copy of David Joyner's Adventures in Group Theory

(4) Prime Numbers and the Riemann Hypothesis

There are many books that I found helpful in high school for number theory, for example this classic by Niven et al.

http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469

Try these books(the authors will hold your hand tight while walking you through interesting math landscapes):

Discrete Mathematics with Applications by Susanna Epp

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

A Friendly Introduction to Number Theory Joseph Silverman

A First Course in Mathematical Analysis by David Brannan

The Foundations of Analysis: A Straightforward Introduction: Book 1 Logic, Sets and Numbers by K. G. Binmore

The Foundations of Topological Analysis: A Straightforward Introduction: Book 2 Topological Ideas by K. G. Binmore

Introductory Modern Algebra: A Historical Approach by Saul Stahl


An Introduction to Abstract Algebra VOLUME 1(very elementary)
by F. M. Hall

There is a wealth of phenomenally well-written books and as many books written by people who have no business writing math books. Also, Dover books are, as cheap as they are, usually hit or miss.

One more thing:

Suppose your chosen author sets the goal of learning a, b, c, d. Expect to be told about a and possibly c explicitly. You're expected to figure out b and d on your own. The books listed above are an exception, but still be prepared to work your ass off.

This one might be good.

http://www.amazon.com/Heres-Looking-Euclid-Counting-Awe-Inspiring/dp/1416588280

This one? How advanced would you say it goes into primes?

I like Underwood Dudley's book, and it's now available cheap from Dover: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?s=books&ie=UTF8&qid=1468275037&sr=1-1&keywords=dudley+number+theory

I recommend http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469. I'm currently reading through it

Fermat's little theorem, Euler's theorem, Euclid's algorithm for greatest common denominators (specifically Euclid's extended algorithm to generate multiplicative inverses), Carmichael numbers, Fermat primality test, Miller-Rabin primality test, modular exponentiation, and discrete logarithms.

If you want to go further you may want to learn about things like finite fields (specifically Galois fields), polynomial rings, elliptic curves, etc. This isn't meant to limit things; e.g., cryptography (and attacks on cryptography) aren't necessarily limited to these types of math. E.g., NUTRUEncrypt is based on lattices/shortest vector problem, and the McEliece Cryptosystem is based on Goppa codes, but again you still need to learn the math above to be able to understand this math.

http://www.amazon.com/Friendly-Introduction-Number-Theory-Featured/dp/0321816196/ref=sr_1_1?ie=UTF8&qid=1425696135&sr=8-1&keywords=A+Friendly+Introduction+to+Number+Theory+%284th+Edition%29+%28Featured+Titles+for+Number+Theory%29

We used this https://www.amazon.com/p-adic-Numbers-Fernando-Quadros-Gouvea/dp/3540629114/ref=mp_s_a_1_1 in a seminar.

https://www.amazon.com/Course-Number-Cryptography-Graduate-Mathematics/dp/0387942939

That's too dificult. Start with some arithmetic.

Sometimes they literally don't know any mathematics, so I teach them some arithmetic.

Get ahold of this book:
http://www.amazon.com/Elementary-Number-Theory-Gareth-Jones/dp/3540761977/ref=sr_1_2?ie=UTF8&s=books&qid=1255891222&sr=8-2

I think An Introduction to the Theory of Numbers was the book I used as an undergrad.

This book is awesome and cheap as hell.

Koblitz is a real red, it's kind of funny. His Course in Number Theory and Cryptography has this dedication:

>This book is dedicated to the memory of the students of Vietnam, Nicaragua, and El Salvador who lost their lives in the struggle against U.S. aggression. The author's royalties from sales of the book will be used to buy mathematics and science books for the universities and institutes of those three countries.

I see how the title could be misleading. One of the most famous examples is Serre's A Course In Arithmetic