Reddit reviews An Introduction to the Theory of Numbers
We found 11 Reddit comments about An Introduction to the Theory of Numbers. Here are the top ones, ranked by their Reddit score.
Oxford University Press USA
For number theory Hardy's Introduction to the theory of numbers is classic. As a more general book for real analysis his Course on Pure Mathematics is also excellent.
I know it is a little out of your proposed areas, but I would also recommend Coxeter's Introduction to Geometry.
All three books are readable, but do require study. However, as they are written by true masters of the areas that study does help develop very deep intuition.
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.
A terrific browsing book in number theory is Introduction to the Theory of Numbers by Hardy and Wright. An oldie but a very goodie.
http://www.amazon.com/An-Introduction-Theory-Numbers-Hardy/dp/0199219869
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.
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.
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.
As I see it there are four kinds of books that fall into the sub $30 zone:
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.
I think An Introduction to the Theory of Numbers was the book I used as an undergrad.
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!
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.
This one? How advanced would you say it goes into primes?