Talk:Number theory

The introduction is a little too focused on number systems, and then mixes them up with all other things. Perhaps we should start with a historical introduction - then an enumeration of the main areas and problems of study? Harald Helfgott 13:55, 18 June 2007 (CDT)

I'm inclined to agree. The initial comment about C.F. Gauss seems out of place in an encyclopedia but, just as importantly, unrelated to the rest of the article. What follows is basically a hodge-podge of ideas presented without any context. In fact, I think it's probably a good idea to just blank the article and start over. A historical introduction may be the way to go, but there are other possibilities, such as outlining some of the main areas of number theory: algebraic number fields, zeta-functions and analytic methods, quadratic forms and lattices (along the lines of Minkowski), p-adic fields and local methods, algebraic geometry (elliptic curves and abelian varieties), and maybe a bit about the Langlands program. Of course, the approaches aren't mutually exclusive: I think Scharlau and Opolka ("From Fermat to Minkowski") does a masterful job of weaving together a historical account and a cohesive theoretical framework. I completely wore out one copy of the book as a grad student. Greg Woodhouse 14:20, 18 June 2007 (CDT)

I have effectively blanked the article. Some text was good and could be reused, but, as it will be available as part of previous versions for at least some time, no harm has been done. Let us see what we can do. Do you want to get started? The overall plan of the Wikipedia article seems sensible. I particularly liked its history section - but then that probably deserves its own article. Harald Helfgott 06:49, 21 June 2007 (CDT)

I have started what should remain a brief history section. Edit away. The modern period has not been done yet. Harald Helfgott 06:43, 22 June 2007 (CDT)

- there should probably be a smooth transition to "Subfields" towards the end of the nineteenth century. Harald Helfgott 08:15, 22 June 2007 (CDT)

There are some problems with the new version of the article. The definition of number theory is incorrect. The study of the integers is called arithmetic, which is one very small part of number theory. You are neglecting the analytic, geometric, topological and computational aspects of number theory if you take that definition.

Analytic number theory is the application of analytical means to the study of integers and their generalisations, alternatively, the study of analytical questions about the primes (for instance). Arithmetic geometry is the study of rational and integer points on varieties, and their attendant structure. As for computational number theory - we are always computing *something*.

I would agree that the stub, as it stands, is exactly that; the label here should be changed. Once the article approaches anything near completion, it will become clear that the integers are often no more than the origin of number-theoretical questions.

The link to Euclid's Elements links to the periodic table not the book.

Ooops.

Although the Chinese remainder theorem was written down in China in the third century CE, I hardly see this as justifying the statement that Chinese mathematicians (plural) were studying remainders and congruences in that period. This is an unwarranted generalisation. We really have no idea what was being studied in that period. In fact it is stated as a problem in the Sunzi suanjing and we have no idea whether a general method was developed around that time, much earlier or not at all.

Thanks! I just introduced a minor change; can you edit that paragraph further? You obviously know more than I do about the subject.

The statement: "In the next thousand years, Islamic mathematics dealt with some questions related to congruences, while Indian mathematicians of the classical period found the first systematic method for finding integer solutions to quadratic equations", is vague and misleading. It may be argued that the Babylonians were able to solve problems which we would interpret as quadratic equations. Some specific dates and names might be appropriate.

I hope matters are clearer now. I should probably write an article about the *chakravala*.

Number theory has historically been motivated by a hodge podge of esoteric problems (at least since the 1500's or 1600's). The original article which has been blanked was written that way on purpose.

The blanking was not meant as an insult. The original article should be used as one of the main materials towards the main section of the present article - unwritten as of yet.

The comment about the origins of modern number theory is highly perspective dependent. Some would say Weber and Hilbert, others would say Erdos, Weil or the Bourbaki group, others would say Euler.

We should probably have the main section on history start with Fermat and Euler, and build up to Gauss.

The article also contains general proofreading errors and is highly incomplete. It is also not well linked with related topics. The article is most definitely not a developing article beyond a stub as classified above.

In general the current article confuses arithmetic (study of the integers, congruences and questions related to primes) with number theory, which has much more to do with number systems.

This is about the one point in which there may be an actual difference of perspective. Number systems are all well and fine, but working with such a system does not amount to number theory. Certainly most of what is done over C is not number theory, and I rather doubt that the number systems used in nonstandard analysis have ever been used in number theory. What makes something into number theory is the kind of questions that are being asked - questions that originated in the integers and can often be formulated in terms of the integers. About the one subfield in which such a statement would seem not to be approximately true is algebraic number theory: while the theory of ideals was motivated by an effort to make algebraic integers behave like the integers, much work done in the subject during the last hundred years has focused on the ways in which algebraic integers do not behave like the (rational)integers -- class field theory, structure of class groups, etc. This is only natural: work is done on areas that present a difficulty. Harald Helfgott 03:25, 9 July 2007 (CDT)

Re the comments above. Algebraic geometry and the Langland's philosophy are not subfields of number theory.

Finally, the Wikipedia article in not a good guide for what should go here. It is tremendously oversimplified and contains numerous inaccuracies as does the article on the Chinese Remainder Theorem. William Hart 07:27, 6 July 2007 (CDT)

I think, that "Problems solved and unsolved" must be only "Problems", and this must be subpage (not subsection). What do you think, about this? Veselin Vavrek 04:02, 8 January 2008 (CST)

I think an essential feature of number theory that separates it from other branches mathematics is the availability of accessible unsolved problems. I believe that this needs to come through in the main number theory page, through a few examples. The article should not purport to give a comprehensive list of unsolved problems, and I agree that a deeper discussion of the types of major open problems deserves its own pageBarry R. Smith 18:48, 20 November 2008 (UTC)

Difficulty Level

I believe that some of the article, as written, is too sophisticated for the number theory main page, being more suitable for an "Advanced" subpage. We should probably discuss what general topics can be discussed on the main page, and which are too sophisticated for it.

This is a thought. At the same time, we are faced with the following issue: we have to describe both what number theory has been historically, and what number theory has been for the last two hundred years. We can discuss Diophantus and Euclid at a level understood by all, but it will be hard to explain what number theory is nowadays without using some technical terms.

One solution - rather than scrapping the "subfields" section - would be simply to complete the historical section, with special attention to the late eighteenth century and the nineteenth century (which I haven't done at all myself yet). This would be a very good point at which to explain, for example, how algebraic numbers were first used and why ideal numbers and ideals were first introduced. (Note, however, that the current "algebraic number theory" section consists largely of exactly such an introduction.)

For instance, I think the average university educated person has trouble grasping the concept of algebraic numbers, probably never having heard the words together before.

You probably mean somebody who received a liberal-arts education. I am not sure that it would do the subject a favour to gear all of the discussion towards former English majors. Rather, we could aim part of the text to, say, somebody who once took some engineering courses (or science courses) and then forgot them, or, alternatively, to somebody who is or was good at maths in secondary school. So - algebraic numbers, yes, that would be new; but complex numbers would be familiar, as would be polynomial equations. I do not see how to do any justice to the topic as it exists nowadays by pitching it to people completely frightened by maths (as opposed to merely ignorant of it, which we all are, to different extents).

As such, it seems to me best just to mention algebraic numbers as being a type of number generalizing the integers as in the introduction, but to keep the discussion of algebraic number theory brief. 

This is too vague.

Certainly, topics in the last two paragraphs or the section about algebraic number theory, namely number fields, field extensions, Galois theory, class field theory, and the Langland's program (!!) are waaaaaaaay too sophisticated even to mention to the average university educated person. 

Why? What harm is there in giving a mere mention to the Langlands program? It happens to be what many people are currently working on. I may not have the background to understand what the large hadron collider really does, but that does not mean it should not be mentioned in the newspapers. The same goes for class field theory

I have added an explanation of what a Galois group is; you are right that I should not have assumed that the reader had ever heard of it. I have added some other explanations to the section, trying to make it more concrete. Tell me what you think of it now (and make your own additions).

This material would be completely lost on them. These topics get so advanced that I would find it hard to argue that class field theory and the Langland's program should even be included on the "advanced" page.

Um. The advanced page would presumably be there for mathematicians and mathematics students who need to get a good overview of a field they are not

themselves familiar with - including a good summary of current activity. How could we do that without giving brief explanations of these topics (as opposed to name-dropping them, which is what we do here)?

Why not include these in a page on algebraic number theory, or even something more advanced than that?

We should probably discuss this once a page on algebraic number theory exists.

I think we can assume that the "typical university educated person" that we are supposed to aim at will have some exposure to calculus of one variable. In the analytic number theory section, it is noted that "analytic methods" refers to calculus-like methods, which is good. But perhaps we can mention that "analytic" refers to calculus-like methods when the word "analytical" is used in the introduction?

That's a good idea.

(Which reminds me, I wish I knew exactly when adding an "al" to words ending in "ic" is appropriate -- geographic/geographical, geometric/geometrical often seem used synonymously as adjectives.  Arithmetic and logic are different, because they function as nouns as well.  I tend to favor "analytic" rather than "analytical", because they seem synonymous to me and I favor brevity.  Any opinions on this?)

Follow current usage. Hence: analytic number theory, but either analytic or analytical in other contexts, depending on what a working mathematician would say.

Again, on the main page, which should be low-level, I don't see why distribution questions about prime ideals in number fields is mentioned. Is the question of the distribution of prime numbers not satisfying enough? Analytic methods are used in the study of that question...

We must mention questions on the distribution of prime ideals in order to show that analytic and algebraic number theory are not two disjoint fields. Analytic number theory studies analytical questions using analytical methods; algebraic number theory studies algebraic objects. Obviously one can do the two things at the same time (i.e., one can study analytical questions on algebraic objects by analytical means). Actually, you are right in that this point should be made more explicitly.

The section on Diophantine Geometry talks about n-dimensional space

Just like 2- and 3-dimensional space, only more so.

, counting the holes in a surface in 4-dimensional space (!!),

What is strange about that? The surface itself is two-dimensional; it just happens to live in 4-dimensional space. Many amateurs know about the Klein bottle, say, which is an example of exactly that.

and genus. I don't think the casual reader will get anything out of this discussion. Why not give a simple example -- say x^2 + y^2 = z^2,

Examples are good. Let me see what I can do.

then mention Fermat's last theorem. Then say diophantine geometry is the study of similar types of problems with more general polynomial equations. The rest of this stuff should be placed on a MUCH more advanced page (in my opinion, it is too advanced even for the main diophantine geometry page).

I cannot possibly agree with what you say within parentheses; see above.

On the other hand, the simplest parts of number theory, "elementary number theory", involving integers and modular arithmetic are not even given a subsection! I think the bulk of the article should be about the history of number theory and elementary number theory, being the most accessible topics.

There is no such field as "elementary number theory". There are subjects which are more accessible to a given individual than others; what these subjects are depends on the individual. These subjects belong to different fields of current research.

There is such a thing as the concept of an "elementary proof" (as in, say, an elementary proof of the prime number theorem) - but an elementary proof may be more difficult to understand (for many) than a non-elementary proof of the same result. By "elementary proof" people generally mean one that does not use calculus, esp. complex analysis; this is paradoxical by now - I think we agree that calculus is precisely the one aspect of university mathematics that a general reader can be assumed to have any familiarity with.

Does anyone agree with me that we should take the entire "subfields" section, move it to an advanced page, and rewrite the whole section from scratch?Barry R. Smith 18:48, 20 November 2008 (UTC)

Well, others may. Right now completing the historical sections I have not yet written is more urgent. Harald Helfgott 22:22, 29 December 2008 (UTC)