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)

Since you wrote most of the history section, let me begin by expressing my admiration for what you have done so far in that area. It will take a while to formulate a complete counter-response, so I will do it in pieces. I think the additions you made today improve what is written. I do believe that class field theory was probably the most important direction of research in algebraic number theory in the first half of the twentieth century, and that the Langland's program appears to be one of the few leading contenders for most important in the last few decades. Perhaps you are right that they should be included on the main page. However, I think we disagree about the utility of name-dropping for informing the interested non-initiate into the nature of our field.

To take your example, the large hadron collider, I believe it is much easier to give a short description of that than of, say, automorphic representations. I believe it is much MUCH easier to give some sense of the importance of what the large hadron collider might produce. It is a really big machine, shaped in a ring. Particles will be sent around the machine at very very high speeds, and a few will smash into each other with higher energy collisions than could otherwise be observed. At several different areas in the ring, observations will be made. Some will give insight to the nature of the physical universe very shortly after the big bang, billions of years ago! We might also observe the Higgs Boson, which will be a coups because it was predicted by theory and is one of the biggest outstanding problems with particle theory.

I guess I could continue on, and maybe I have a detail or two wrong, but the point is, anyone can understand the summary I just produced, because I am a layman myself. Anyone can understand "really big machine". I think explaining "adele", or "automorphic representation" is qualitatively different. Without that basic explanation and a semblance of its importance, I don't see the point of name-dropping.

Many people can get a good idea of Fermat's Last Theorem without having to be exposed to "all elliptic curves are modular". Sure, you can just write the words and link to a page on the modularity conjecture, by name-dropping, but I think it ruins the sense of unity in the page (unity in sophistication of content, here). This is even though any decent explanation of the history of Fermat's Last Theorem will talk about this fact.

I can see the point of name-dropping on the advanced version of a page. I am not sure of the intent, as you suggest, that an advanced page in the mathematics category should be aimed at least at budding mathematicians. I have been taking my cue from the quantum mechanics page and the quantum mechanics/Advanced page for the appropriate level of sophistication for a page. Even as a non-physicist, and someone who has read barely any physics during the last ten years, I can understand most of the Advanced version of the page. I think even the frightened "english majors" can understand virtually the entire quantum mechanics main page. I think achieving this level of exposition for such an esoteric subject is quite an impressive feat, and I was assuming that these are models of articles aimed at their intended audience. The impressive name-dropping, words liked "quantum gravity", "string theory", and "quantum field theory", is all done on the advanced version of the page.

As far as the current state of the algebraic number theory section, I still think the average person won't have any idea what "abelian group" means, and will have trouble getting a quick understanding. I like your example of complex conjugation in the description of Galois theory, but I doubt that the short description of field automorphisms would make sense to the layman. Perhaps it makes sense enough, but I don't know. And "field" is still mentioned on the page -- that can be succinctly described in a good paragraph, so if the word is used on the main page, I think it should have that descriptive paragraph. I understand that my opinions are not those of a layman, so we really cannot know how the section reads until a layman weighs in.Barry R. Smith 01:49, 30 December 2008 (UTC)

OK - three things.

(a) I think we agree that we want to keep name-dropping to a minimum. We don't want this to be NUMB3RS. Still, it is sometimes unavoidable. We are talking about one of the major endeavours in contemporary mathematics; it is difficult to think of one of the same scale and reach with any relation to number theory. Not mentioning the Langlands program would make this article defective in so far as the sociology of mathematics is concerned. If, say, we were writing an article about knighthood, the Holy Grail were beyond all explanation, we would still have to say that some guys were after it (preferably under the appropriate subdivision of "subfields of knighthood").

The quasi-explanation of the Langlands program as being a generalisation of class field theory to the non-abelian case may not be completely satisfactory, but at least it is standard.

(b) In the advanced version of the page (once we get to that - this page isn't even close to finished), we would give more technical explanations, and omit some of the simplest ones. Eventually, hyperlinks would take care of undefined terms. This is one of the advantages of working with hypertext; this is not a print encyclopaedia. (I just thought of explaining "abelian" in a footnote, but I think I'll add a link instead; we already have an article on "abelian group". I'll add a sentence to that article so that it's clear what an abelian group of maps (automorphisms) is.

(c) The point of the general page should be to introduce the topic to as broad an audience as one can reach without distorting the topic grossly. We should always explain what can be explained without fear of condescension. At the same time, it seems to me that readers should be prepared not to understand everything. We should omit not that which is not understood by somebody, but that which is understood by almost nobody. Of course, we have to try to make things as understandable as possible (though not more so).

As for the advanced page - let us take a look at Wikipedia. (I know it is not necessarily a good model; more about that in a second.) As a mathematician, I use it quite a bit; I am sure you do too. Many of the technical articles are quick, accurate and to the point. They are useful because they can be used by a category of people to which such articles are actually of great use - namely, mathematicians (physicists, CS people, etc.) who want accurate information quickly on a field generally not their own. It seems fair to me to aim at the same audience with articles on more advanced topics, as well as "advanced" articles on more general topics.

Ah yes - general topics, such as this one. By general consensus (in maths departments, Wikipedia talk pages, your average pub, etc.), Wikipedia articles on general topics in mathematics suck dead rabbits through a bent straw. (I hope I am not breaking any civility rules here.) They are vague, inaccurate, out of touch with current research, and generally written by uneducated laymen and hobbyists armed with fifty-year old popularisations - there is often somebody with a bee in his bonnet as well.

So, when it comes to non-"advanced" articles on general subjects, it seems to me that we are not to follow Wikipedia's example. At the same time, there is a large niche left open if we write an advanced page with (a very broad class of) professionals in mind. This niche consists of articles written by mathematicians (like you and me) for the public of all those who are mathematically interested (otherwise, why come to this page?), have some vague mathematical literacy and are willing to put some effort (the same would go for a page on philosophy, say), but cannot be assumed to have taken any third- or fourth-year courses in mathematics, say. I think we can assume calculus (when needed) and a willingness to look up basic concepts in abstract algebra elsewhere in citizendium (given proper links).

We better get some guinea pigs. My (anthropologist) brother read the article, but he is not necessarily unbiased. Harald Helfgott 03:05, 30 December 2008 (UTC)

Algebraic number theory

I've reworded to make it clear that this is, as I believe, the algebraic theory of numbers, not just the theory of algebraic numbers. Richard Pinch 10:29, 1 January 2009 (UTC)

You are right. Quadratic reciprocity and Fermat's result on the sum of two squares have been in some sense assimilated into algebraic number theory, yet (in their original formulation) they are non-trivial statements about the rational integers. Harald Helfgott 15:41, 1 January 2009 (UTC)

Hence my confusion about the statement, "there is no such thing as elementary number theory". It may be a neologism, but the phrase is fairly common today, as evidenced by a quick search for books called "elementary number theory" on Amazon. Presumably there was no reason to make such a distinction until a century or so ago. Much of elementary number theory is a special case of results about algebraic number fields, so perhaps can be absorbed into "algebraic number theory" with Richard's usage. Some topics, like the study of special types of primes, Fermat/Wilson quotients, Perfect/Amicable numbers, or the Collatz conjecture don't seem to me to be readily lumped into one of the more advanced subfields.

As elementary number theory is the most accessible subfield, and has the broadest interest for "hobbyist" number-theorists, I still think it should have its own section -- the first one on the list.Barry R. Smith 18:34, 1 January 2009 (UTC)

With all due respect - the existence of books entitled "elementary number theory" does not prove the existence of a field called "elementary number theory" any more than the existence of books called "elementary gardening" would prove the existence of something called "elementary gardening".

I'll try to write the sections on Fermat and Euler soon; I hope to make them completely accessible. Perhaps they will even help make the material in the later subsections more accessible. Still - part of early modern number theory was assimilated into algebraic number theory; some of analytic number theory can be done without complex analysis; additive combinatorics can be quite accessible - and all of that accessible material belongs in the section of each subfield. Harald Helfgott 18:53, 1 January 2009 (UTC)

You are right. But the existence of two books by different people called "elementary gardening" and discussing roughly the same sort of gardening techniques would prove the existence of a field called "elementary gardening". A word or phrase has meaning when two different people both use it and agree what it stands for. Now if these two hypothetical authors were the only people in the world who used the phrase "elementary gardening", it would be considered an extremely rare usage, and hence would be inappropriate for use on a Citizendium page. My contention is that "elementary number theory", if not common usage, is approaching it.

You have not explained your aversion to the phrase. If you find the phrase distasteful, for some reason, consider this: most people find many racial slurs to be distasteful, and very rarely hear such words in their social circles, but you will still find the major slurs in most major dictionaries. Distaste is not a reasonable excuse for exclusion.

Is it because you prefer an alternative term? In my understanding, "elementary number theory" usually refers roughly to results, usually about rational integers or numbers, some about modular arithmetic, that require no advanced knowledge to state and often do not require advanced techniques to prove. Perhaps you think of such material as the "classical" theory of numbers, and so will be subsumed under the historical section when it is fully developed? I do not agree with this -- there is current research being done on elementary results, often still using elementary (but often clever) techniques.Barry R. Smith 20:42, 1 January 2009 (UTC)

It seems to me that "elementary number theory" is in part an artifact of pedagogy. There seems to be a mainline of twentieth-century number theory books - possibly all descended from Vinogradov. (This is a guess; I don't have any sources for this.) Before, in the nineteenth century, you had Lucas's book (most of which wouldn't be called number theory nowadays) and then you had Dickson's book (a lot of which is spent on forms, if I remember correctly; it's really a very different choice of topics from Vinogradov). There are also mid- to late- 20-th century books that don't follow Vinogradov's basic outline: the main example here is Hardy and Wright.

There's also the concept of "elementary proofs". In the context of number theory, this basically means proofs that do not use complex analysis - especially if such proofs replace previous proofs that do use complex analysis. Elementary proofs, in this sense, are often less enlightening and harder to understand at any level than the "non-elementary" proofs they replace, though in principle they require less previous knowledge. (Much of this knowledge is of a kind that mathematics undergraduates in any subfield regularly get.) I strongly doubt that anybody who does not know any complex analysis has ever actually read Selberg and Erdos's proof of the prime number theorem, for example. (Of course, I have no way of showing that.)

Then there's the issue of proofs being done nowadays with what look like pretty elementary means. Actually, such proofs are often done by people with a thorough knowledge of many subfields in modern number theory, and they are motivated by ideas that are not elementary. (Some of the work of A. Granville crosses my mind here, for instance.)

What about the following? Let us have two main sections in the paper: History of Number Theory, and Number Theory nowadays. The second subsection can consist of a short subsection and a long one. The long subection would (for the while being) be what is now the "Subfields" section. The short subsection would come before the long one and be entitled "Introductory topics and elementary proofs".

Perhaps what I have just written can serve as some sort of very rough draft for the first version of the short subsection: we could talk about first courses in number theory, about elementary proofs, and about so-called elementary work being done nowadays (emphasising, perhaps in the indirect way I did it above, that almost all of the interesting things are done by people who know much more than the elementary stuff). Of course, all of that would have to be more NPOV and less opinionated than what I just said. What do you think? Harald Helfgott 23:27, 1 January 2009 (UTC)