# Talk:Number theory

Many questions in elementary number theory are exceptionally deep and require completely new approaches. Examples are

What's that supposed to mean? Of the four conjectures, three haven't yet been proven. Mihailesca's proof of Catalan's conjecture has been said to use only elementary methods (and it has been said that the newer versions of the proof use no mathematical theory beyond what was available to Catalan himself).

As for the other three, I wouldn't rule out elementary proofs of any of them, and it is possible some of them cannot be proved at all. It's still not clear any of them would require completely new approaches.

Prumpf 15:26, 23 Nov 2003 (UTC)

I agree with the above. It doesn't make any sense to say they are exceptionally deep when they are very simple concepts, and it doesn't make any sense to claim that they require completely new approaches when we simply don't know.

Mysteronald 18:49, 5 Sep 2004 (UTC)

It does make some sense to say that they may lie deep or require new methods (which is not quite the same, but something closely related). Obviously simple assertion falls down on the test of being POV.

Charles Matthews 21:27, 5 Sep 2004 (UTC)

Sure. That does make some sense. This is a mathematical topic, and I think there's value in being precise in this instance. I've changed it, then.

Mysteronald 12:41, 6 Sep 2004 (UTC)

Combinatorial number theory should not be omitted! I hope someone could write description of this important branch of number theory.

Larry Hammick 07:44, 4 Apr 2005 (UTC)

Maybe the article should mention Diophantine approximation as a branch of number theory.

But it does, under analytic number theory. I would regard combinatorial number theory as growing out of ANT, also, but I can see that not everyone would agree. Charles Matthews 10:57, 4 Apr 2005 (UTC)

## Elementary?

In the section of "Elementary Number Theory", it mentioned four difficult problem, three of which is still open. I won't agree that these problem should put into this section, just because they're appear to be simple. When people're talking about elementary number theory, Goldbach problem will never come into their mind. I hope someone could make a better arrangement for these topics. —Preceding unsigned comment added by Chinmin (talkcontribs) 14 November 2005

The "Elementary" in Elementary Number Theory means, as stated in the article, that no results from other fields are used; not that the subject is simple or obvious. Erdos' proof of the Prime Number Theorem was elementary, but obviously not simple or obvious. Contrast this to things like Analytic Number Theory, which uses results from analysis; and Algebraic Number Theory, which uses results from abstract algebra.Phoenix1177 (talk) 10:01, 2 January 2008 (UTC)

## Jargon as opposed to development of thought

Guys (and gals), this "article" is a fine, bulletized shopping list of topics within number theory, but it completely lacks any sort of thought development...it is something of a technicolor yawn of topics, with nothing connecting the jargon.

In has no character of lucid explanation. There is no development of thought, and it merely reads like a jargon-laden index (only), not an article. It's something of a number theory "jargon router," and needs to be either entirely re-written to show a flow of thought, or to have someone make the considerable effort to connect the many, many jargon "dots" that have been thrown down on the floor.

As it exists, it is an excellent list, but a poor explanation. No offense intended.

It might be salvageable if someone were to take the time to (1) slow down, rather than skipping merrily from one jargon-label to another, and (2) exhibit some sense of both connectedness and branches of thought. --AustinKnight 12:15, 11 December 2005 (UTC)

Well, number theory doesn't really exist in the kind of unified way that supposes. Various branches do hang together. Therefore something of the kind can be considered inevitable. The only way to mollify the bittiness is to write more of the history, but that assumes much of the reader. Charles Matthews 12:25, 11 December 2005 (UTC)
That doesn't get past the problem of "shopping-list-ness." The article is virtually opaque in terms of thought-development. It would help to provide some basis (brief definition?) as to why "(jargon-label-of-choice here)" has something to do with number theory. If there is no connectedness, how can there be an overall label (number theory) for the topic? --AustinKnight 12:29, 11 December 2005 (UTC)
Number theory is notoriously the hardest part of mathematics. The article is called 'number theory' because that is what the topic is called. The lack of 'connectedness' is characteristic of all so-called combinatorial mathematics; in other words areas driven by the type of problems to solve, rather than the methods used to solve them. What I see is that the history section is badly incomplete. Charles Matthews 12:36, 11 December 2005 (UTC)
The fundamental problems that I'm seeing with the article are that it is too terse and cryptic...something that a mathematics undergrad or grad student might find useful for routing-of-interest purposes, but there's not much use beyond that. It certainly isn't encyclopedic. --AustinKnight 12:37, 11 December 2005 (UTC)
I do agree that fixing the history section would help fill-in-the-blanks when it comes to development of thought. That'd be a fine approach for fixing the article, which needs to be a tree with branches, rather than a pile of leaves on the ground. --AustinKnight 22:49, 11 December 2005 (UTC)

What about application of number theory to computer science? Maybe to put some links to other relevant articles of application to CS.--Čikić Dragan 14:45, 1 February 2006 (UTC)

## Picture?

Isn't that picture a little off-topic? Elliott C. Bäck 02:25, 10 March 2006 (UTC)

An attractive picture displaying numbers would be unnecessary. This is worse. Nor do other wiki branches of mathematics have a representational illustration (thankfully) MotherFunctor 23:53, 16 April 2006 (UTC)

I was horrified to see such an 'illustration'. It does not belong to a serious article. I am going to cut it immediately.

a pkture tells more then a 1000 words. bring the pictur back NOW. you silly woman eliot--194.237.142.10 09:12, 20 June 2006 (UTC).

## Cleanup tag

I asked Stevertigo to explain why he placed the cleanup tag on this article. Here is his explanation, copied from my talk page ... Gandalf61 08:08, 7 April 2006 (UTC)

The second paragraph on the etymology is out of place - get into the general areas what number theory is about instead. Is it limited to integers? Is there no number theory about complex numbers, etc? They way the intro is stated it makes it seem as if NT is a misnomer for integer theory. Its just not clear on the generalities ... Its an editorial issue, not a math issue - be explanatory. -Ste|vertigo 01:06, 7 April 2006 (UTC)

## Relevant mathematicians

I find there are several people named in the sections on 19th and 20th century number theory whose contribution to number theory is quite unclear to me. For example, Poinsot, Lebesgue, Borel, Tannery, Schering, Glaisher, Genocchi. Should these names really be here? Both of these sections need a lot of work. For example, class field theory is generally accepted as being a rather important development in 20th century number theory. RobHar 05:51, 31 May 2007 (UTC)

how about some (spacial) asian number theory, with all its (curious) numbers and (novel) theories? 04:07, 7 August 2007 (UTC)04:07, 7 August 2007 (UTC)~~

It's hard to tell from your non-standard terminology but maybe you think of numerology and numbers in Chinese culture. That is not considered number theory which is a mathematical discipline. Similarly, astrology is not astronomy. PrimeHunter 05:00, 7 August 2007 (UTC)

I am a number theorist, and my first reaction to this page is to propose that it be deleted. This is probably an over-reaction, but still I find it unhelpful and almost embarrassing. What is gained by these extremely brief, often not very insightful descriptions of topics that are (or could be) described in much more detail in other articles? Surely anyone who is curious would learn more by going to a page listing number theory topics and clicking on one of interest?

Here are some specific criticisms:

(i) The heading "elementary number theory" is confusing. Is it reporting on the subfield of contemporary research (practiced, e.g. by Granville and Pomerance) that straddles the "low tech" part of analytic number theory and the burgeoning field of combinatorial number theory -- both areas in which the arguments are characterized by getting good estimates on various quantities -- or is it describing a sophomore level university course?

The fact that there are easy to state and hard to solve questions is not characteristic of number theory. As has been noted above, reporting that certain unsolved problems "may require very deep considerations" is not false and not even really POV, but it is completely unhelpful. Also why have these five problems been chosen?

How is the algorithmic undecidability of Diophantine equations a part of elementary number theory?

(ii) Some indication that the first sentence describing analytic number theory is just a guideline describing classical work would be nice -- nowadays a large portion of analytic number theorists study properties of L-functions which are at some remove from the properties of integers. (Again the fact that L-function is not mentioned here makes me think that whoever wrote this article does not know enough about number theory to speak for it.)

Transcendence theory is really its own branch of number theory, and should be treated as such.

(iii) "The virtue of the machinery employed [lots of words dropped]...allows to recover that order partly for this new class of numbers."

First, I don't understand the sentence: what order? Unique factorization? What recovery process do you have in mind?

Here you see my point: dropping terms like Galois theory, class field theory etc. with such little context provided (and of course it would be very difficult to write an article providing proper context for all of these things) is no more informative than just listing these topics on a page, but it creates a worse impression.

Reducing modulo p is called "reduction"; completing with respect to a prime ideal is called completion or localization. The field of study is _not_ called local analysis by anyone I know.

(iv) Similarly I have never heard of the term "geometric number theory"; when I saw it, my guess was that it was supposed to describe arithmetic geometry, and was disappointed. (Where is arithmetic geometry in this taxonomy?) To say that it incorporates "all forms of geometry" is strange: it would be nice to think that, say, the Ricci flow is used by number theorists and some day it may well be so, but (of course) not all aspects of geometry are currently being incorporated into number theory, the same as with any other two different fields of mathematics.

(v) "Combinatorial number theory deals with number-theoretic problems which involve combinatorial ideas..." Again, please defend the use of such sentences. (It does go the other way around as well: you can use number theory to solve combinatorial problems, e.g. Ramanujan graphs.)

I won't comment on the history -- it's not worth it.

Plclark 22:45, 19 August 2007 (UTC)Plclark

Wow. I am sure no one will argue with your view that this article is far from perfect. The original authors of the specific statements that you take issue with are probably far away in the mists of Wikipedia, so I don't think you will get answers to many of your questions - although most of them seem to be rhetorical anyway. If you think the article should be improved then the best and simplest way to achieve this is to improve it yourself. Minor changes to wording and terminology can be made directly into the article - the worst that can happen is that someone disagrees with your change and reverts it. Major changes, rewrites, restructuring or removals of whole sections should be proposed on this talk page first - this is considered good wikiquette, it gives folks the chance to comment, and can save wasted time and effort. Finally, if you feel that the article is so fatally flawed that it cannot be saved then you can formally propose its deletion at Wikipedia:Articles for deletion. Gandalf61 11:31, 20 August 2007 (UTC)
This article is definitely extremely lacking, and at times quite dubious. Other than the content (which definitely needs some attention) the layout itself should be improved. For example, the section on algebraic number theory appears word for word in the wiki article Algebraic number theory. I would suggest that each branch of number theory that is big enough to have its own wiki article should have one of those main article templates indicating the relevant main article, and in the section in this article there should be a sentence about the beginnings, the main object(s) of study and maybe a major result and/or current trends of research. What do people think? RobHar 05:59, 28 August 2007 (UTC)
Concur: ... completely with RobHar. There's no reasonable objection to the structural enhancements, none that I can see. dr.ef.tymac 14:32, 28 August 2007 (UTC)

## Jain math

I don't think that the section on "Jaina mathematics" should remain in the article.

1. I don't think it's mathematical so much as philosophical. The different classes don't seem to correspond to actual infinite numbers.
2. Most of these are actually the same: the first three would apparently all be countable, by the article description, so it they're actually infinite they're all ${\displaystyle \aleph 0}$. Space-filling curves show that "infinite in one dimension" is the same as "infinite in two dimensions" and so for any finite number of dimensions -- though I imagine "infinite in infinite dimensions" would be distinct. There seems to be no mathematical method in listing these, just some sort of philosophy.
3. The section is unreferenced.
4. I wasn't able to reference the section. sacred-texts.com doesn't have the book, nor does Project Gutenberg. WorldCat library search can't find either "Surya Prajinapti" or "Surya Prajnapti". Google pulls up mostly Wikipedia mirrors and a lot of sites clearly copying from Wikipedia.
5. The development of modern cardinalities is unrelated to the Jain mathematics, and (as mentioned above) the two don't have a clear mapping between them.
6. Number theory has less to do, in general, than most branches of math with the various sizes of the infinite. If this section has a home, I wouldn't expect it to be here.

So is there something I missed that would make this useful?

CRGreathouse (t | c) 03:07, 27 March 2008 (UTC)

I agree; this doesn't seem related to number theory. The proximate source for this material is most likely the MacTutor Archive article on Jaina mathematics. Ben Standeven (talk) 18:37, 18 April 2008 (UTC)

## History

I do not think that the Vedic section belongs in the article. There are no citations, and some of the claims seem unrelated to number theory. There was, of course, later Indian number theory - starting with Aryabhata and going from then. By the way, shouldn't we have a section on Babylonian mathematics? The tablet with "Pythagorean" triples arguably belongs here. Feketekave (talk) 13:11, 8 May 2008 (UTC)

## "Geometric number theory"

Shouldn't we have a section on Diophantine geometry (or its highbrow cousin, arithmetic algebraic geometry)? Feketekave (talk) 15:42, 27 May 2008 (UTC)

## "Algebraic Number Theory"

In algebraic number theory, the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. [...] In this setting, the familiar features of the integers (e.g. unique factorization) need not hold. The virtue of the machinery employed—Galois theory, group cohomology, class field theory, group representations and L-functions—is that it allows one to recover that order partly for this new class of numbers.

what "order" exactly are we talking about? Zero sharp (talk) 22:17, 2 December 2008 (UTC)

I believe "order" refers to the "familiar features of the integers". But this sentence certainly could benefit from some rewriting. It is also present in the lede of the algebraic number theory article, which itself needs to be improved. RobHar (talk) 02:28, 3 December 2008 (UTC)

## Cluelessness

OK, I confess that I'm feeling a bit cranky. I'm just tired of encountering and trying to clean up tons of WikiNonsense in edits contributed by people who really ought to be aware that they don't have the remotest idea what they are talking about. Case in point, from article:

"Carl Friedrich Gauss conjectured the limit of the number of primes not exceeding a given number (the prime number theorem) as a teenager."

The "limit of the number of primes not exceeding a given number" ??? Puh-leeze.Daqu (talk) 08:11, 1 April 2009 (UTC)

## Number theory as pure math

I think there's need to be a lot of evidence before the wiki article should start claiming number theory is not a branch of pure mathematics. Some applications of modular arithmetic and a bit of elliptic curves does not "change all that". Analysis firmly remains a branch of pure math despite calculus being a rather applied form of mathematics. RobHar (talk) 16:22, 5 August 2009 (UTC)

I don't think it is very useful to use "pure" in the definition of large areas such as number theory or analysis which have many applications. I'm not a great fan of the distinction between pure and applied. However I've been reverted by two editors so I guess I'm out-numbered. Charvest (talk) 15:29, 6 August 2009 (UTC)
It is my opinion that number theory is the study of numbers, particularly things derived from, analogous to or related to the integers. In this sense, its pursuit is within the realm of pure mathematics. That some of its methods, results, or theories can be applied to problems outside of mathematics does not contradict its nature as pure math. I believe it is a rather uncontroversial statement to say that the vast majority of number theorists engage in the study of pure mathematics, thus making number theory a branch of pure mathematics. I say this because I don't believe that my opinion out-numbering yours is sufficient to decide the content of this encyclopedia. RobHar (talk) 16:05, 6 August 2009 (UTC)
I don't dispute that Number Theory and other areas of mathematics are done mostly from a pure point of view. I just don't like defining them solely as pure. I prefer to just call it mathematics and leave it at that. Charvest (talk) 16:58, 6 August 2009 (UTC)
I agree with the majority here that it would be considered a branch of pure mathematics. I have no preference between linking to Mathematics and Pure mathematics. CRGreathouse (t | c) 04:43, 7 August 2009 (UTC)

## Possible proof for Collatz conjecture

(Post moved to mathematics reference desk and deleted by poster)

This talk page is intended for discussions of the contents of the number theory article. As your post is more general than that, a better place for it would be the mathematics reference desk. Also, you may get more useful responses if you explain you proof strategy in English, before giving the formal version. Gandalf61 (talk) 08:42, 2 October 2009 (UTC)
OK, thanks.--Gilisa (talk) 09:57, 2 October 2009 (UTC)

## Misuse of sources

Template:User-multi is one of the main contributors to Wikipedia (over 67,000 edits; he's ranked 198 in the number of edits), and practically all of his edits have to do with Islamic science, technology and philosophy. This editor has persistently misused sources here over several years. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. Please see: Wikipedia:Requests for comment/Jagged 85. The damage is so extensive that it is undermining Wikipedia's credibility as a source. I searched the page history, and found 13 edits by Jagged 85 in April/March 2006. Tobby72 (talk) 19:38, 15 June 2010 (UTC)

## Restart?

I do think this page is in need of serious work. (See the section "Please explain the need for this page".) I would be in favour of starting again. Some time ago I started writing a page on Number Theory for Citizendium (a parallel project for Wikipedia): [[1]]. I wouldn't mind at all if it were taken as a basis for a newer article, though I suppose Citizendium policies would have to be consulted.

By the way, of course, I am a number theorist. Garald (talk) 15:35, 4 July 2010 (UTC)

Additions and extensions to Wikipedia articles are always welcome (provided, of course, that they conform to our policies and guidelines). But I think you would need to provide a substantial justification for re-starting this article from scratch. Gandalf61 (talk) 17:11, 4 July 2010 (UTC)
Sounds fine. Check on the licensing issues and start it at User:Garald/Number theory. When you get it to a presentable state, post here with a link and we can discuss replacing or merging the articles. CRGreathouse (t | c) 18:33, 4 July 2010 (UTC)
Very well. The licensing is fine - I've started things at User:Garald/Number theory, and would welcome feedback. Incidentally, how do I get confirmed (so that I can upload images)?
As for reasons why the current page is bad: the history is choppy and amateurish at parts, and the same can be said about the division into fields. Moreover, it feels as if most of the text had not been written or structured by a practicing number theorist. There are bits of good information, but it would make more sense to graft them onto a better structure, rather than using the existing structure as something onto which to graft information from elsewhere. That's my personal opinion - but it seems opinions like that are very widespread. Garald (talk) 16:52, 5 July 2010 (UTC)

Right now I'm on a long academic trip, and thus away from (most of) my library. It would be good if, during this lull in work on User:Garald/Number theory, I could get some feedback (on the talk page here or there) on the shape it is taking. Garald (talk) 11:19, 14 August 2010 (UTC)

Back at work now. Garald (talk) 17:42, 9 November 2010 (UTC)

I'm sorry, but as a non-number-theorist, I find User Garald's page much less useful than the current "official" page. It seems to be more appropriately titled "History of Number Theory", and he is talking to himself and perhaps to other number theorists. The current article by contrast provides a useful overview with pointers to more detailed articles on subdisciplines. If number theorists wish to pat each other on their backs, Garald's version may be what they need. If Wikipedia wishes to provide an authoritative entry point into the current subject for nonspecialists, please leave the current article alone, or merely correct it where it is wrong or amplify it where it could be made more clear. I suspect that there are far more non-number-theorists in your audience than specialists, and we need an article that is genuinely helpful, not pedantic. Thanks! 96.233.20.134 (talk) 06:17, 20 September 2011 (UTC)
I think the first thing to point out is that Garald's version is a work in progress and in its current form is not yet meant to replace this page. Secondly, if by "a useful overview with pointers to more detailed articles on subdisciplines" you mean the content of the section "Fields", well that content is quite meager in most cases (e.g. "See arithmetic geometry"), and presumably the finished form of Garald's article will cover these fields and more while still providing the pointers to more detailed articles you mention. Finally, number theorists, and certainly Garald, don't need to take time out of their busy lives to "pat each other on the back"; rather they are writing wikipedia articles in the interest of other people. So, if you have some constructive criticism of Garald's version then by all means go to the talk page of his write-up. RobHar (talk) 14:09, 20 September 2011 (UTC)

## "The Higher Arithmetic"

The introduction says that "the higher arithmetic" is used to refer to elementary number theory. Is this correct? As part of a taught Masters I took a course called "Higher Arithmetic" which was all about analytic number theory and the prime number theorem. The Arithmetic article says that "Professional mathematicians sometimes use the term (higher) arithmetic when referring to more advanced results related to number theory…", the article contains a source. The inclusion of the phrase comes from a single book: "The Higher Arithmetic: An Introduction to the Theory of Numbers." This title seems to be from a popular science point of view, and not a true mathematical point of view. I shall remove the the phrase "the higher arithmetic" from the introduction, whilst warmly welcoming any discussion regarding the future inclusion of the phrase. 22:35, 2 July 2011 (UTC)

I just want to say that the book The Higher Arithmetic by Davenport is not a "popular science" book, but an extremely well-respected textbook by an extremely well-respected number theorist. Furthermore, whether or not "higher arithmetic" refers to elementary number theory, it certainly is not reserved for analytic number theory. I do think that "higher arithmetic" refers to any form of number theory at least as advanced as elementary number theory (see this where Gauss uses the term). RobHar (talk) 22:49, 2 July 2011 (UTC)
Added: In fact, in the foreword to the Disquisitiones, Gauss defines the term "Higher arithmetic" (a term which he decided to expand as evidenced in the above link). There are several results on google books that elaborate on the term. I'm confused as to why you jumped on removing this term from this article. RobHar (talk) 22:56, 2 July 2011 (UTC)
Agree with RobHar. I have got Davenport on my bookshelf. It is definitely not "popular science", nor is it about analytic number theory. As Wolfram says, "higher arithmetic" means number theory in general. I have reverted the removal. Gandalf61 (talk) 08:27, 3 July 2011 (UTC)
I've got a brief explanatory text on this at the beginning of [[2]] - a draft of a proposed new version of this page; I intend to return to it now. Garald (talk) 12:03, 17 August 2011 (UTC)

## Restart

The new version is almost ready. I've moved most of the historical part to a separate page, which I am now completing. Please comment on the talk page there. Garald (talk) 13:55, 7 October 2011 (UTC)

By the way, I would like to thank the people who commented in some of the sections above ("Jargon as opposed to development of thought", "Please explain the need for this page", "Restart?") for narrowing down what is wrong with the current page and what some of the things wanted from a new page would be. Garald (talk) 14:36, 7 October 2011 (UTC)

Is "Peano arithmetic" really parallel with "floating-point arithmetic"? (Has the latter been formalised in a similar way?) Garald (talk) 15:24, 10 October 2011 (UTC)

I think this is very similar: both define a representation of some numbers (integers for Peano, a finite subset of the real numbers for floating point. Both define operations on them. For a long time the result of a floating point operation was not uniquely defined, because correct rounding was not defined. But now the floating point operations are well defined by the norm IEEE 754. By the way another example of this use of arithmetic is computer arithmetic. D.Lazard (talk) 16:01, 10 October 2011 (UTC)
Point taken. However, perhaps the last paragraph of the introduction offers too much detail extraneous to the subject by now? We could always start a section on "naming" at the end of the article - a detailed discussion of (a) the different names for number theory, (b) the different meanings of 'arithmetic' would belong there, and we could just keep a briefer explanatory note at the beginning. Garald (talk) 17:46, 10 October 2011 (UTC)

## "Too many opinionated sentences"

Speaking as a number-theorist here. While Garald's new version isn't bad, here are some comments.

I think it has too many sentences that read like opinions. I think that in an encyclopedia, opinions should be either omitted or be rephrased and backed up with citations. Here are some of the sentences I am talking about: "Any attempt to conduct such a study naturally leads.." "Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods." (This sentence should be omitted entirely, in my opinion)

Also, when I was learning number theory in my olympiad days, I found the book of Burton extremely useful, and I think it should be added to the list.

Finally, how about adding automorphic representations to the section of recent subfields? (with some comments on analysis on adeles /Tate's thesis and modular forms) — Preceding unsigned comment added by 129.132.146.66 (talk) 10:01, 11 October 2011 (UTC)

Hello IP-editor -

Thanks for your comments. If you don't get a username, everybody will start trying to guess which number theorist currently in Zurich you are. :)

I appreciate that you mention a book I didn't know, since I can now look it up - but we better keep the list to established classics, or otherwise its length will become subfields.

Please do add more on automorphic representations - under analytic number theory, I would say. Garald (talk) 10:28, 11 October 2011 (UTC)

For what is worth - the comment on Hardy and Wright is cited and referenced. Adding Serre's Course in Arithmetic to the list might be worthwhile (though it assumes more background than the other two), but then we should really close the list. Garald (talk) 11:18, 11 October 2011 (UTC)
I agree with anon - too many opinions on many things. While I personally agree on most of them, they are inappropriate here. The literature choices are potentially contentious. While I like Hardy-Wright, and much of the q-series they discuss is very helpful to my students, I doubt many number theory grad students can even recognize this to be NT these days. Similarly, Vinogradov's book is essentially an intro to analytic methods - while important, better and more accessible books exist now. In general, this article has to be seriously edited to bring to WP:GA standards. Igorpak (talk) 01:16, 12 October 2011 (UTC)
Then go ahead and edit. It's a good sign that people are now willing to put in the effort.
Just to answer your questions - the literature choices were meant to be (a) classical (so as to avoid having an infinite list of everybody's personal favorite), (b) helpful to people who want to get started - that's the purpose of this page. If the consensus really is that this does not belong elsewhere, then so be it, but we better discuss it. (I personally feel that the course of action should be to tone down the comment on Hardy and Wright, and simply put Serre's book as a third alternative, saying it requires quite a bit more background in other branches of mathematics - whereas Hardy and Wright hew to elementary (arguably non-analytic) proofs of analytic statements.)
I don't know whether exactly how many number theory grad students recognise q-series as part of number theory; if they don't, it just means they have something to learn. I think we have quite a few editors now that have gone quite a bit beyond grad school. Incidentally, the one criticism that can be fielded here is exactly what I meant: nowadays people who get to a certain level in number theory can be assumed to know complex analysis, and hence part of what Hardy-Wright treat in an elementary fashion with formal power series can be done more sleekly with complex analysis.
Lastly, I don't think things that are to some extent judgement calls (other examples?) are necessarily bad in and of themselves, as long as (a) they are supported by a consensus (preferably of specialists, or, if not, at least of everybody who knows things in depth), (b) they are couched in NPOV language. Perhaps it would be best to compare this page not to, say, the page on q-series (technical and uncontroversial) but to the page on any other large subfield of any field of study. There are going to be some appraisals as to what is and what is not central, as well as to what is and what is not history; there's that thing going on even in the article on the circle method, say (to give a random example I've just looked at). Garald (talk) 08:33, 12 October 2011 (UTC)
PS. Good call on moving that to the literature section. Garald (talk) 08:45, 12 October 2011 (UTC)

## Classical vs. recent

I am not sure that I like the new headings ("classical approaches and subfields" vs. "recent approaches and subfields"). The "classical" fields are still the main fields, i.e., the fields with most development and activity. They are very much current. It's not clear that the "recent approaches and subfields" are all on the same footing: a) probabilistic methods in number theory are of well-established usefulness, but they have to be seen either as a substantially smaller subarea or as a collection of tools for attacking problems in one of the big areas; (b) "arithmetic combinatorics" is a very recent denomination for fields that were seen as disparate, and to be honest still are to a large extent - I have no confidence as to whether people will still speak of additive or arithmetic combinatorics fifty years from now; (c) algorithms are a very important thing, but is the study of their complexity a branch of number theory, or a branch of complexity theory with number theory as its object of study?

Perhaps we should try not to go further into that - rather, it would be better if some clever person came up with a form of organising this that did not imply a clear parity or parallelism. Garald (talk) 08:44, 12 October 2011 (UTC)

Yeah, I made titles as placeholders - feel free to change them or unify two sections into one. In Combinatorics I did make a distinction between areas in the field and borderline areas in other fields. I think it's a useful distinction, which allows to write more about core NT and less about applications. Igorpak (talk) 15:36, 12 October 2011 (UTC)

## Pictures

Also, can we come up with some pictures? Perhaps a colorful phase graph of the Riemann zeta function for the analytic section and some cyclotomy for the algebraic section, for instance. The example given under 'diophantine geometry' could certainly use a graphical explanation - a (generic-looking projection onto 2 dimensions of) a generic-looking projection onto 3 dimensions of a four-dimensional (over the reals) plot of ${\displaystyle \scriptstyle y^{2}=x^{3}+ax+b}$ for some values of a and b, say. Garald (talk) 09:02, 12 October 2011 (UTC)

Not a bad idea. There are many already available, like this one for zeta function. Some pages like Collatz conjecture already have many good pictures - just walk around the field and collect maybe 4-5 which you can include in the page. Igorpak (talk) 15:56, 12 October 2011 (UTC)
What is the protocol for adding images from outside Wikipedia? I'd like to add a picture of a Lehmer sieve - this one [[3]], for instance. What about copyright and the like? Garald (talk) 13:44, 14 October 2011 (UTC)
Um, it's all in WP:IMAGE and WP:FIT. Right. For copyright reasons, essentially you can't download an image from the web and add it to WP, with minor technical exceptions which are not applicable in this case. You can create the image yourself or request one - somebody might be willing to make it. Igorpak (talk) 22:40, 15 October 2011 (UTC)
Went and asked in the Village Pump. Let us see. Garald (talk) 11:38, 20 October 2011 (UTC)

## Pictures redux

Found and added a properly licensed picture of a Lehmer sieve. It is, of course, extremely sexy. I resisted the urge to put it at the beginning of the article, since factorizing large integers and finding numerical solutions to simple diophantine equations are not actually what we number theorists do with our time. On the other hand, we also don't use mechanical devices with gears and beams of light, and that somehow doesn't strike me as a reason against the picture. What do you think?

Likewise - it should be easy to make a picture of an application of Minkowski's theorem to illustrate the algebraic number theory section; I've hesitated so far simply because the geometry of numbers is now arguably a tiny part of algebraic number theory - smaller than it used to be, anyhow. Here I'm leaning to include a picture anyhow. Garald (talk) 09:45, 29 October 2011 (UTC)

## What is to be done?

We have two tasks facing us: putting this article in shape so that (a) it becomes a Good Article, (b) it becomes a good article.

(We may start by acknowledging that these tasks need not be exactly the same; at times they may be orthogonal or worse.)

Discuss. Garald (talk) 09:45, 29 October 2011 (UTC)

You may already know this, but GA review has a reputation in the Wikipedia mathematics community of being a rather hostile environment, hence the very small proportion of mathematics articles that are GAs (25 articles and 5 bios) and the low frequency of GA nominations for mathematics articles(none at the moment). A few pointers based on my own experience of GA nominations (2 passes, 3 fails):
1. Before even thinking about a GA nomination, this article needs a lot more references. GA reviewers like to see three or four references per paragraph - see Penrose tiling to get an idea of what is required - and this article is currently very sparsely sourced from "Main subdivisions" onwards.
2. Do not underestimate the amount of work involved in getting an article through a GA review. If you are lucky, you will get a very lengthy and detailed GA review that lists several dozen points that need addressing. If you are unlucky you will get a short GA review that says "I didn't understand this article - take it away and rewrite it from scratch".
3. Be aware that most GA reviewers expect an implicit commitment from nominators that they will address any and all points raised by the reviewer in short order and without argument. They will be very critical if you take too long to address their points, suggest that they are incorrect, or abandon the nomination. Don't make a GA nomination unless you are absolutely sure you have the time and patience to follow it through to the end. Gandalf61 (talk) 15:07, 29 October 2011 (UTC)
Some more references would be nice. At the same time, what you say, in general, makes me suspect that the GA review, at this stage, may simply not be well geared towards mathematics articles. The main reason why there are "enough references" before Main Subdivisions and not thereafter is that the section before Main Subdivisions is a history section. In a history article, every statement can and should be referenced. In a mathematics article, we are talking either about universally true statements ("1+1=2") or about statements that - given enough edits - have or will become statements that essentially every mature mathematician agrees with, but that are hard to reference ("1+1=2 is easy", amended to "1+1=2 is relatively simple to prove in most systems, not including that of 'Principia Mathematica'"). It is very hard to see how to write a good article (as opposed to a Good Article) without these two statements.
Hence, perhaps we should discuss further how to improve this article, without having a GA nomination in mind (and thus without necessary reference to GA norms). A-class may be a different matter, since assessment is inevitable (or isn't it?). Garald (talk) 10:49, 31 October 2011 (UTC)

(On the other hand, how far do you think History of number theory is from a state where it would be good to nominate it for GA status? Garald (talk) 11:27, 1 November 2011 (UTC))

- Also, do you think History of number theory should really be merged with this article? I originally conceived the two as one article; I may have gone too far in splitting it. Garald (talk) 14:59, 11 November 2011 (UTC)

I have reintegrated the longer version of the history section into the article. (If this holds, I'll later delete the separate History of Number Theory page, which has received few edits.) I have also added a few references at the end. It is my opinion that section 2 is now up to the point where the entire article should be if we are to nominate it for Good Article status. There are, in particular, plenty of references.

Now we have to work on the rest. Does anybody else want to take over particular subsections of "Main subdivisions" or "Recent approaches and subfields"? Garald (talk) 19:17, 14 November 2011 (UTC)

## Re-format of refs and citations

In response to Garald's above question "What is to be done?", I added a bibliography section, and re-formatted all the cites and notes to use Harvard referencing and Shortened footnotes. This system uses the {{sfn}} and {{harv}} or {{harvnb}} templates in conjunction with the |ref=harv parameter of {{cite book}} to automatically build and link (multiple) references to the same book and page. >MinorProphet (talk) 00:35, 15 April 2012 (UTC)

Kudos for this! Garald (talk) 15:58, 17 April 2012 (UTC)

## Diophantine approximation and Transcendence theory

I was rater surprised that the word "transcendence theory" and "transcendental" did not appear in the article before my recent edit and that the few words on Diophantine approximation were rather difficult to find at the end of "Diophantine geometry". Should we create one or two separate sections for these two areas? An element supporting that is that Diophantine approximation is now, IMHO, a decent article. D.Lazard (talk) 15:56, 2 July 2012 (UTC)

I think that (as you hint) if a new section is to be created for the two, it should be one section, and not two separate sections; the two areas are closely intertwined.

I'd say that the connections of the two areas with Diophantine geometry are strong enough that they can all go under one section, but that is a matter of taste. (Notice that, say, Roth's theorem is included in Hindry and Silverman's Diophantine Geometry.) At any rate, the least one can do is add a link to "Diophantine approximation" under "Main articles" after the Diophantine geometry section title. Garald (talk) 14:11, 30 July 2012 (UTC)

As other people have mentioned before, there's also a pressing issue with the article - we have to come up with citations for all of the "Main subdivisions". This is tricky; since this is a survey section on a very large field, some of the choices and descriptions will have to rest on the judgement of professionals editing this article. What would be some good, recent, detailed survey works we can cite? The survey in the Britannica (for example) is well-written but old, and its choices of subjects to highlight is - in my view ' decidedly outdated. Garald (talk) 14:11, 30 July 2012 (UTC)

For transcendence theory and diophantine approximation, the historical section of the article by Waldschmidt, cited in Diophantine approximation seems a good reference. D.Lazard (talk) 18:49, 30 July 2012 (UTC)
In case you decide to use the above ref, it appears to be an extract from the following (with other, newer sections):
* Template:Cite web
* e.g. {{sfn|Waldschmidt|2008|pp=71-78}} HTH, >MinorProphet (talk) 21:28, 30 July 2012 (UTC)

## Subfields

If we are to request to be rated and be confident to be raised to A-class - or if we want good article status - we need many more citations from "Analytic number theory" downwards. For this to be possible, we would need to rewrite a fair bit of the article; otherwise we are playing a difficult game of get-the-citation.

What would be some useful sources for citations? I doubt most textbooks or handbooks would be very useful, unless they have substantial historical sections or undertake a broad survey at first.

Perhaps each subsection can be structured as follows - a definition, a somewhat historical narrative, and a discussion of solved and unsolved problems on the side or afterwards? The history of individual problems is easy to document (with citations to the original works if necessary). Thoughts? Garald (talk) 19:47, 15 August 2012 (UTC)

## Andre Weil

Andre Weil's slanging Fermat reflects Weil's own failure on the same point. — Preceding unsigned comment added by 78.105.0.33 (talk) 13:20, 28 August 2012 (UTC)

Andre Weil also slanged perfect numbers, which he had also failed at. — Preceding unsigned comment added by 78.105.0.33 (talk) 14:07, 3 September 2012 (UTC)

In section 2.3, the last sentence of the first paragraph, is not readable to me:

If ${\displaystyle A+A}$ is barely larger than ${\displaystyle A}$, must ${\displaystyle A}$ have plenty of arithmetic structure \pmod e.g., does it look like an arithmetic progression?

Since it contains some unusual notation (\pmod), and English is not my mother tounge, I cannot repair it myself.
--H.Marxen (talk) 19:28, 11 January 2013 (UTC)

I just rewrote that sentence, so it should be more readable now. — Anita5192 (talk) 21:43, 11 January 2013 (UTC)

## Literature

The claim regarding the 2 most popular introductions seems a bit dubious to me. In any case I think it would be helpful to mention at least a few more recent introductory textbooks (say for instance Niven/Zuckermann/Montgomery) and the advanced literature on number theory could have a few more titles as well.--Kmhkmh (talk) 05:03, 13 February 2013 (UTC)

I went ahead now and added 2 more recent popular introduction texts (each with 5+ editions). I think for more advanced texts we should ideally offer a book for each subdivision or subfields mentioned in the article. However since I'm not really versed in those, I'd prefer somebody else to add those to assure the choice is appropriate and not just matched/guessed by title.--Kmhkmh (talk) 15:51, 17 February 2013 (UTC)