Background needed to understand number theory research papers.

[owowo]

Hi,
I am a computer science student that has found an interest in mathematics. I am currently exploring number theory, among other fields such as abstract algebra, and have gathered an interest in it after glancing at HAKMEM and Hacker's Delight, as well as learning of its importance in fields such as cryptography and quantum computing. I would like to eventually reach a level where I can start comprehending number theory research papers and even start my own research (all through self study, of course). I have started working through the following books:
- An Introduction to the Theory of Numbers 5e- Niven, Zuckerman, Montgomery
- Number Theory - George E. Andrews
and I have also ordered:
-An Invitation to Modern Number Theory - Miller, Takloo-Bighash
I want to know if these books will provide me with enough background on number theory to understand today's number theory research papers or carry out my own research. If not, would someone kindly inform me what else would be needed to do so (books, papers, topics, links...etc will be useful)?

Thank you in advance.

P.S As a note, I think it would be nice to have a sticky in each section providing this kind of information to amateurs in the future.

 

[owowo]

Anyone?

*sorry for bump

 

[rasmhop]

(I'm not myself capable of doing research in number theory yet, but I have some knowledge of what it takes as I'm considering specializing in it. Take my advice with a grain of salt as I may not really have the authority to speak on this matter.)

It depends on the kind of research. Number theory is a huge field. After reading and fully understanding the three number theory books you may be able to understand a couple of research papers on number theory (mostly old ones), but to be blunt 3 books will not be anywhere near enough to do serious research in modern number theory. In fact most of the contents of these books can be covered in an undergraduate elementary number theory course. Number theory is a kind of peculiar field and even today sometimes elementary proofs of elementary results are published, but most of the researchers doing this have a firm foundation in mathematics and number theory from where they draw inspiration.

Mathematics students use 7+ years of full-time study guided by professors before they are able to take on original research, and even then it's usually guided by a professor. Just like there is no royal road to geometry, there is no royal road to number theory. You have to learn it the hard way and that takes years of dedication. Mathematics is highly interconnected and you cannot just ignore large parts of it.

Number theory is a diverse field and you do not need to understand it all, but if for instance you want to understand algebraic number theory you should have a decent grounding in algebra (at least at the level of a serious graduate book like Lang's Algebra), and a couple of graduate books on algebraic number theory (algebraic number theory by Lang is a good introduction, and most books by Jean-Pierre Serre are good follow-ups). This should prepare you to study the prerequisites for a more specific area for which you will need a couple of more graduate level books (maybe you want to focus on certain kinds of cohomology in number fields, or global class field theory). Similar roads are needed for other kinds of number theory. For instance in analytic number theory you will need a solid grounding in analysis as well as a couple of graduate books on analytic number theory.

Of course while reading these books you will find that to understand them, and more importantly to understand their motivation, you need to read up on other mathematical topics such as Galois cohomology, diophantine geometry, algebraic topology, homological algebra, etc.

Once you have done all this you may be able to understand the technical details in number theory papers, but of course you will not have the experience yet to perform original research. For this you will likely need more perspective from talking to active researchers, reading articles, and studying related parts of mathematics.

To give you an extent for the prerequisites let me quote from "Algebraic Number Theory" by Serge Lang which is supposed to be an introduction to algebraic number theory:

Chapters I through VII are self-contained, assuming only elementary algebra, say at the level of Galois theory.
Some of the chapters on analytic number theory assume some analysis. Chapter XIV assumes Fourier analysis on locally compact groups. ...

Galois theory, and Fourier analysis on locally compact groups are pretty heavy prerequisites for most CS students. Especially in an introductory text.

You will basically need the equivalent of a phd in math to actively do research in number theory. I'm not trying to discourage you as math and number theory are beautiful subjects worthy of studying even if you do not reach the research level, but assuming those books are enough is like assuming reading "An introduction to algorithms" (CLRS) is enough to do independent modern CS research in the field of algorithms. It's only an introduction that prepares you for much deeper treatments (but CLRS is enough to understand many applications of algorithms outside of CS).


You should know however that modern number theory isn't what is used in most computer science applications. Most of the computer science applications are of elementary number theory of the type covered in Niven. Even when more advanced number theory is used it's usually in the form of a theorem that looks elementary, but where the only known proofs uses advanced number theoretic machinery.

Consider that pretty much every graduate student in mathematics has the knowledge equivalent to what is obtained from your books, and they are all looking for potential research topics. In addition they have professors to help guide them, and who themselves are looking for worthwhile research to do. And these people have a lot more related knowledge and perspective (which helps a lot in math). You may spot what they do not and come up with an elementary proof for a remarkable result, but the odds are not good until you get to much more specialized fields and gain knowledge comparable to the other researchers.

Most of the low-hanging fruit has already been grabbed in number theory as it's such an old subject, and the only fruit (problems) left needs very elaborate arguments and heavy machinery. Of course once in a while someone stumble upon a single fairly elementary problem, but it becomes rarer as the subject matures and more people have looked for these problems.

 

[owowo]

Thank you rasmhop for that lengthy reply!
I just want to make sure, when you say:

Consider that pretty much every graduate student in mathematics has the knowledge equivalent to what is obtained from your books, and they are all looking for potential research topics.

do you mean that when students first enter their graduate program, they start off with the material I am currently studying? If so that is a comforting thought :)

I think you have given me some good advice on specializing in one topic of number theory. I was confronted with the same decision when I saw the huge number of topics in AI.

Now the question is, what should I specialize in?
I would have to think this through. To tell you the truth, I would really like to take a shot at an open problem, something to keep me thinking and reading to further my mathematical knowledge, but I do not know the respective field of each problem. I probably do not want to come up with a problem (ie research question) as a PhD degree might require, but I certainly would want to look at something that people have puzzled over for a while. I would appreciate any suggestions on topics that might be interesting, but it is not necessary at this point.

Thanks.

 

[rasmhop]

do you mean that when students first enter their graduate program, they start off with the material I am currently studying? If so that is a comforting thought :)

Yes a lot of them will. And those that don't will have the necessary background to pick it up easily.

Now the question is, what should I specialize in?

That is not really something you need to answer now. You do probably not know what goes into studying the cohomology of number fields, or even what exactly algebraic number theory and analytic number theory are. You need some fundamentals before you can specialize. Books such as Niven are great for getting you into elementary number theory and after reading that you will have a much better idea of what is to come, and what you really like. I sometimes hear high school students talking about how they want to do string theory or nanotechnology, but they do not have the necessary knowledge to be able to fully understand what these subjects entails. In a similar manner you need to work on your foundations and as you progress it will always be clear what the possible roads ahead are.

To tell you the truth, I would really like to take a shot at an open problem, something to keep me thinking and reading to further my mathematical knowledge, but I do not know the respective field of each problem.

Well you can take a look. Plenty of open number theory problems are easily understandable however you will likely not solve one in the immediate future. The reason for this is that thousands, if not tens of thousands, have tried the same problems. If there were a fairly easy open problem, then someone would solve it as all mathematicians look for open problems they can solve. The only ones left are the ones research mathematicians have a hard time with (or haven't thought of yet, or that aren't interesting).

Also there is not necessarily a particular field of a problem. This is not a problem set given by your professor. The problem is open because no one knows how to solve it. You may use techniques from all over mathematics. Algebraic geometry, algebra, functional analysis, etc. can all be used in number theory proofs. This is also one of the reasons why you can't specialize too early. You will not have the necessary set of techniques to tackle a problem.

I have seen at least a couple of books (mostly popular mathematics aimed at people with limited background) that try to explain some easily explainable open number theory problems, but remember that all the others that bought the book have looked at these problems and thought about them so the obvious approaches won't work as they have been tried out.

Again I do not mean to discourage you, but it really takes a lot of studying before you're able to do serious research in mathematics. You'll likely not get a real problem under your belt for quite some time, however I don't see what is wrong with just working with solved problems (such as the ones given in your textbooks).

 

[Petek]

Since you're a computer science student, you might be interested in computational number theory. See the Wikipedia article for some references.

HTH

Petek