(Part 2) Reddit mentions: The best number theory books

We found 205 Reddit comments discussing the best number theory books. We ran sentiment analysis on each of these comments to determine how redditors feel about different products. We found 56 products and ranked them based on the amount of positive reactions they received. Here are the products ranked 21-40. You can also go back to the previous section.

21. Elementary Number Theory

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22. Number Theory: A Lively Introduction with Proofs, Applications, and Stories

Number Theory: A Lively Introduction with Proofs, Applications, and Stories
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24. Asimov On Numbers

Asimov On Numbers
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25. An Introduction to Number Theory (The MIT Press)

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26. Concise Intro Theory of Numbers

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27. Elementary Number Theory with Applications

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32. Elementary Number Theory

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33. Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften (322))

Algebraic Number Theory (Grundlehren der mathematischen Wissenschaften (322))
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37. An Invitation to Modern Number Theory

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38. Algebraic Number Theory

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39. Trolling Euclid: An Irreverent Guide to Nine of Mathematics' Most Important Problems

Trolling Euclid: An Irreverent Guide to Nine of Mathematics' Most Important Problems
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40. Invitation to Number Theory (New Mathematical Library)

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🎓 Reddit experts on number theory books

The comments and opinions expressed on this page are written exclusively by redditors. To provide you with the most relevant data, we sourced opinions from the most knowledgeable Reddit users based the total number of upvotes and downvotes received across comments on subreddits where number theory books are discussed. For your reference and for the sake of transparency, here are the specialists whose opinions mattered the most in our ranking.
Total score: 104
Number of comments: 24
Relevant subreddits: 1
Total score: 68
Number of comments: 3
Relevant subreddits: 2
Total score: 32
Number of comments: 6
Relevant subreddits: 2
Total score: 31
Number of comments: 5
Relevant subreddits: 1
Total score: 15
Number of comments: 5
Relevant subreddits: 1
Total score: 10
Number of comments: 5
Relevant subreddits: 1
Total score: 9
Number of comments: 5
Relevant subreddits: 3
Total score: 7
Number of comments: 2
Relevant subreddits: 1
Total score: 6
Number of comments: 3
Relevant subreddits: 1
Total score: 2
Number of comments: 2
Relevant subreddits: 1

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Top Reddit comments about Number Theory:

u/Goku_Mizuno · 2 pointsr/learnprogramming

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.

u/blaackholespace · 18 pointsr/math

> Mathematical Logic

It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.

Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.

Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.

If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.

Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc

This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.

Last, but not least, if you are poor, peruse Libgen.

u/functor7 · 8 pointsr/math

Intoduction to Langlands Program you can find downloadable pdfs online. This is a fair overview of everything, not many rigorous proofs, but with something as ambiguous as Langlands the general idea is what is important.

To be able to tackle it, you're going to need to know pretty much all math up to second year graduate courses, especially Measure Theory, Representation Theory, Group Cohomology and Harmonic Analysis. You will then need, in addition to that, Class Field Theory and Tate's Thesis. These can both be found in Cassells and Frolich, though a more readable Tate's Thesis is also in the first book. This is just the standard reference book for Class Field Theory, I would recommend looking at Primes of the Form x^2 +ny^2 first. It is written at the advanced undergrad/beginning graduate level, but it is a very excellent read (probably my favorite book) and gives clear motivation for Class Field Theory which is where other books usually fail. It's good to have other sources too, like Neukirch or Lang.

TL;DR You need all of math to talk about Langlands.

u/brandoh2099 · 3 pointsr/math

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.

u/xanaxmonk · 3 pointsr/learnmath

While the hs and early college courses are critical; i found it more helpful to start with one of the more accessible proof based topics: such as elementary number theory (pdf), graph theory or combinatorics.

I like this strategy more because you might find these a lot more interesting, which helps a lot with the most important part of studying: consistency. Additionally, I find starting over a bit demoralising – and many algebra skills can be improved upon while working on something a lot more captivating, anyways.

If you tell me your interests, I can recommend you more specific beginner takes on certain topics.

u/[deleted] · 2 pointsr/Libertarian

It's funny because "reality" is another word like that.

Next, you'll claim "reality" for the progressives because they have "science", which is incidentally another one of those words.

Almost everybody is in agreement in almost all cases: For example, I think we'd agree that abstaining from PCP is "rational", that "reality" contains more conscious beings than just you and your cat, and that physics is a "science" (but even there, the jury isn't 100% in re: string theory).

Physics seem to make pretty decent predictions. If you doubt physics, you're free to go stand inside a fusion reactor as it powers up to become the hottest place in the solar system.

Compared to physics, do you think keynesian economics, gender studies and cultural anthropology has an equal claim to the word "science", thus defining "reality", which in turn defines "rationality"? Do they require the same amount of faith?




u/MrCompletely · 3 pointsr/askscience

Three Roads to Quantum Gravity is a book on this subject for the layperson, by Lee Smolin of the Perimeter Institute. Smolin has written further on the subject, and is considered a strong critic of string theory particularly after the publication of The Trouble With Physics, and in turn has come in for considerable criticism himself. Many string theorists seem to consider his views unworthy or ill-founded, but then, they would.

Another critique of string theory is Not Even Wrong by Peter Woit

I found all of the above interesting, but then I find practically all well-written scientist-authored physics books interesting (not that large a sample size really). All a layperson can hope to do in a situation where experts disagree is to consider as many educated opinions as possible and keep an open mind. So I do recommend the above as interesting but can't speak to their merit as an expert would.

u/_SoySauce · 2 pointsr/math

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.

u/JonahSinick · 2 pointsr/math

It's worth mentioning the books by Kato, et. al:

Number Theory 1: Fermat's Dream

Number Theory 2: Introduction to Class Field Theory

Number Theory 3: Iwasawa Theory and Modular Forms

They're awfully nice. I recommend them as a starting point over most references that have been mentioned.

Feel free to email me at jsinick (at) gmail (dot) com if you'd like to discuss.

u/madmathfuryroad · 3 pointsr/math

If you're a math hobbyist without a lot of formal experience, I think that it'd be difficult to access most books. Apostol is definitely really good, as well as all the suggestions others are making. Strayer is great for someone without a lot of math experience, I'd recommend that one.

u/puzzlednerd · 2 pointsr/math

It might be fun for you to start with an elementary number theory book, such as this one. Number theory is incredibly rich and deep, and pretty much any area of math has been touched by number theory in some sense. The problem is that it's hard to learn math in isolation. It helps to know some other people who care about learning math. You can of course keep posting here, but I'm curious - do you know anybody in real life who shares this interest?

u/For_USA · 1 pointr/VirginiaTech

I graduated In Aerospace Engineering with a minor in math 2 years ago. The professor, Rogers (if he is still teaching) is really good in office hours. I Would also recommend having a study group in that class. We had a group of 4 spiting ideas off another till we understand the material.

this book helped : https://www.amazon.com/Introduction-Fourier-Integrals-Dover-Mathematics/dp/0486453073

If math 4425 is still Fourier Integrals

I hope this helps. good luck

u/Lhopital_rules · 64 pointsr/AskScienceDiscussion

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

u/nullbot · 1 pointr/math

I like An Invitation to Modern Number Theory. A little less readable (imo) but Diaconus had a pretty good article that he published a while ago.

I'm no expert, but from what I understand, Random Matrix Theory is primarily concerned with the eigenvalues matrices whose entries are drawn from some distribution. For many 'natural' distributions, you find that the eigenvalues follow a sem-circlular-law. The five-second explanation of this is that, since eigenvalues are the roots of the characteristic polynomial of your random matrix, and you're in some sense drawing random characteristic polynomials, roots of said polynomial tend to 'repel'. How they repel, what that looks like, etc. etc. is the basic gist of a lot of this work.

u/gmartres · 3 pointsr/math

Finite calculus is awesome, but it doesn't even have its own Wikipedia article. A Primer of Analytic Number Theory has the first chapter devoted and with more details than Gleich's tutorial (for example, the relation between the harmonic function and the logarithm)

u/TheRationalZealot · 1 pointr/ReasonableFaith


You said….

>It makes no sense to demand causation outside of spacetime.

And then….

> I guess someone should tell all those physicists to stop wasting their time.

The second statement appears to follow from the first.


>In what conceivable way has quantum gravity been disqualified?


Can you explain (like I’m a 4-yr-old) what you mean by quantum gravity in regards to the origin of the universe? Is this the theory where the origin of the universe is a freak quantum fluctuation?


>String theory isn't even testable yet, and somehow one of the most active areas of research in theoretical physics is just ruled out?!


I don’t know much about this theory, but it’s not falsifiable which means it takes faith to believe it. Plus this doesn’t answer the questions on where the branes came from. You still end up with an infinite regress of causes, which is impossible. If you can accept a non-falsifiable claim, then you may want to re-examine your motives for not believing in God since string theory is merely replacement theology.


>Oh yeah, and eternal inflation hasn't been "debunked" either, but whatever.


Yeah, it has. BGV did this.


>Not only do BGV leave room for an eternal universe (which is a common view among cosmologists)


Not for our universe or any other expanding universe. Vilenkin subscribes to the multi-verse theory where we are a freak quantum fluctuation. The irony is that the laws of physics that describe our universe have to pre-exist the formation of the universe in order for the universe to form at all. How do you have laws of physics in place before the origin of the universe? How do you have gravity with no mass?


Let’s say the multi-verse is full of quantum foam for creating other universes. Then it becomes a literal Hilbert’s Hotel since in an infinite amount time a universe will form in an infinite number of locations and still have room for more universes. The multi-verse concept when carried out to its logical conclusion becomes absurd.


>Vilenkin's own work describes the universe beginning by way of uncaused quantum tunnelling... how's that for inconvenient?


Are you sure it’s uncaused or is it indeterminate? A source would help, because I don’t think anyone is pursuing a causeless origin. If there was no cause, then how can there be an explanation? Only those who believe the anthropic principle is adequate say “we’re here, so why ask?”.

u/acetv · 3 pointsr/math

You should check out the books published in the New Mathematical Library series.

Here are a few I think might be really awesome:

Geometric Inequalities by Kazarinoff

Invitation to Number Theory by Ore

Numbers: Rational and Irrational by Niven

Mathematics of Choice: Or, How to Count Without Counting by Niven

Episodes from the Early History of Mathematics by Aaboe

Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger

I wish I knew about these books when I was in high school.

u/lbgator · 1 pointr/math

I really like Asimov On Numbers. Amazon usually has a few copies that cost less than $5. It's one of the few books that I've read multiple times.

Edit: I suck at links. Also noticed that other peoples suggestions are better than mine :)

u/doubtingapostle · 11 pointsr/math

I taught myself elementary Number Theory out of his book A Concise Introduction to the Theory of Numbers. It's a pretty dense little book, probably a little too intimidating for most beginning students unless you're willing to expound upon it, but it's nice to have a fairly dense number theory course that fits in a coat pocket.

Sorry to hear of his passing.

u/BeetleB · 3 pointsr/math

Number theory problems are the most common in Project Euler.

Here's the book I learned from. It's good for the intro level. There may be cheaper good ones out there...

u/brokenspirometer · 1 pointr/math

When I was I a kid I enjoyed Asimov on Numbers. It's very basic and very readable.

u/user371820576 · 4 pointsr/math

There's loads of good entry-level number theory books. I would seriously recommend this one

It looks short but covers quite a lot considering that including constructing a transcendental number and the group law on elliptic curves.

u/mathwanker · 1 pointr/math

I like Solved and Unsolved Problems in Number Theory by Daniel Shanks. It takes a unique approach, showing how particular problems led to the development of number theory.

For a more "standard" approach I like An Introduction to Number Theory by Harold Stark, which was the textbook used in the course I took as a sophomore.

u/EricTboneJackson · 0 pointsr/atheism

> Not really. For every religion's claim about objective reality you can find a dozen about morality.

First of all, bullshit. I've read the Bible and Qur'an (and parts of the Vedas, et al). They make a handful of moral claims (however often they are repeated), but these are overwhelmed by chapter after chapter of claims about people, places, events, origins/causes, the nature of the Universe, etc.

Second of all, most religions claim that morality is objective, coming from the same source as the Universe itself, which is a claim about objective reality.

> a lot of people don't think stuff like "existence of God" is about objective reality, that can be investigated scientifically.

First of all (again), bullshit. Show me the theist who thinks the existence of god is a matter of opinion, a matter of taste, something that exists only in your head.

Second, that something cannot be investigated scientifically does not mean it's not a claim about objective reality. For instance, string theory or interpretations of quantum mechanics are often held to be non scientific because they cannot be tested. That doesn't mean they aren't claims about objective reality. There are finite limits to what is testable about the objective world, which place finite limits on science. That doesn't make them subjective.

We don't know what happened before the big bang, or what lies beyond the event horizon of the Universe, and as far as we can tell, we will never know, we have no way of knowing. That doesn't mean science has subjective answers to these questions, it means it has no answers to these questions.

> For example Sam's opponent in that debate holds that position.

No he doesn't.

> The debate was about morality and values, the words before and after this fragment were about values

Right. So if the fragment was about the speed of light, that would make the speed of light a matter of subjective opinion. After all, the words before and after it were.

Jesus.

> How about "Sound is the universe making love to my ears"? I say it's an answer and a subjective one. You can't really say if it's right or wrong.

That's what I just said. You're talking about how sound makes you feel, which is totally subjective.

Unless you meant this literally, an objective claim that the Universe is a sentient animal which is in romantic love with you and is having physical intercourse with your ear canal using it's Universe Sex Organs, in which case: WTF?

u/47Ronin · 3 pointsr/INTP

Anxiety attacks are a serious issue. See a professional.

Also, buy a used copy of Asimov on Numbers and read it. I DARE YOU to panic about math when reading this book.

This book didn't teach me how to do advanced math, but it did make me feel like less of an idiot when I engaged with it, and made the subject in general more approachable.

http://www.amazon.com/gp/offer-listing/0517371456/ref=sr_1_1_olp?ie=UTF8&qid=1411679006&sr=8-1&keywords=asimov+on+numbers&condition=used


u/zakk · 5 pointsr/Physics

Truth is that more and more physicists are starting to dispute the validity of string theory.

http://xkcd.com/171/
http://www.amazon.com/gp/product/0465092756

Even though, I wouldn't kind define it as "bullshit", is simply well-founded theory a theory which strives for finding some kind of verification.

Edit: spelling.

u/PhdPhysics1 · 2 pointsr/AskScienceDiscussion

There was a time when String Theory was viewed as promising, but I think that era has past. These days, a large percentage of Physicists view ST as failed and a cautionary tale about what happens when science becomes decoupled from experiment. There are lots, and lots, and lots of books about this topic.

​

I like to think about things as follows... As far as we know, ST is the only consistent way to unify the 4 fundamental forces while quantizing gravity. This unification requires multiple dimensions, super symmetric particles, and a negative cosmological constant. Unfortunately, Dark Energy is in direct conflict with a negative cosmological constant, super symmetry is looking less likely, and LIGO has found no evidence of extra dimensions. So if anything, ST is strong evidence that a Grand Unification Theory does not exist, and perhaps a new approach is needed. I know many Physicists realize this (perhaps not publicly but at least privately). This is why at the bleeding edge of research we are seeing forays into new areas, e.g. emergent gravity from quantum information, space-time from entanglement, etc.

u/WG55 · 6 pointsr/Christianity

I have a book you might be interested in: Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law. In physics, there are many scientists who have lost the ability to distinguish metaphysics from science.

u/ieattime20 · 3 pointsr/democracy

>Ignorance isn't a crime, willful ignorance is

Are you not willfully refusing to investigate a philosophy you're criticizing?
>Richard Dawkins can explain the basic premise of atheism without sending someone to read a book

...and convince precisely no one, and arm those he does or doesn't convince with absolutely no ammunition to deal with the issue in the future.

I read books like that precisely because it clarifies where the problems of such systems are. And there are problems, but they are subtle and must be couched in the proper context to actually have a discussion. You hear something about String Theory implying that there's giant infinitely dense fibers spanning the universe, and you say "That's absurd." Do you really think you would be equipped at this point to argue with a top-grade physicist, even if some other top-grade physicists think the theory is wrong?