Talk:Number theory
Number theory has been listed as a level3 vital article in Mathematics. If you can improve it, please do. This article has been rated as BClass. 
WikiProject Mathematics  (Rated Bclass, Topimportance)  


Wikipedia Version 1.0 Editorial Team / Vital  (Rated Cclass)  


Contents
 1 Untitled
 2 Elementary?
 3 Jargon as opposed to development of thought
 4 Picture?
 5 Cleanup tag
 6 Relevant mathematicians
 7 how about
 8 Please explain the need for this page
 9 Jain math
 10 History
 11 "Geometric number theory"
 12 "Algebraic Number Theory"
 13 Cluelessness
 14 Number theory as pure math
 15 Possible proof for Collatz conjecture
 16 Misuse of sources
 17 Restart?
 18 "The Higher Arithmetic"
 19 Restart
 20 "Too many opinionated sentences"
 21 Classical vs. recent
 22 Pictures
 23 Pictures redux
 24 What is to be done?
 25 Reformat of refs and citations
 26 Diophantine approximation and Transcendence theory
 27 Subfields
 28 Andre Weil
 29 Unreadable sentence
 30 Literature
 31 This article is somewhat lacking...
 32 Algorithmic undecidability
 33 Citation #1 not understood
 34 Integers, or natural numbers?
 35 Assessment comment
Untitled[edit]
 Many questions in elementary number theory are exceptionally deep and require completely new approaches. Examples are
 The Goldbach conjecture concerning the expression of even numbers as sums of two primes,
 Catalan's conjecture regarding successive integer powers,
 The Twin Prime Conjecture about the infinitude of prime pairs, and
 The Collatz conjecture concerning a simple iteration.
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?[edit]
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 (talk • contribs) 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[edit]
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 jargonladen index (only), not an article. It's something of a number theory "jargon router," and needs to be either entirely rewritten 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 jargonlabel 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 "shoppinglistness." The article is virtually opaque in terms of thoughtdevelopment. It would help to provide some basis (brief definition?) as to why "(jargonlabelofchoice 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 socalled 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 routingofinterest 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 fillintheblanks 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.User:Vanished user 8ij3r8jwefi 14:45, 1 February 2006 (UTC)
Picture?[edit]
Isn't that picture a little offtopic? 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 eliot194.237.142.10 09:12, 20 June 2006 (UTC).
Cleanup tag[edit]
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. Stevertigo 01:06, 7 April 2006 (UTC)
Relevant mathematicians[edit]
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[edit]
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 nonstandard 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)
Please explain the need for this page[edit]
I am a number theorist, and my first reaction to this page is to propose that it be deleted. This is probably an overreaction, 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 Lfunctions which are at some remove from the properties of integers. (Again the fact that Lfunction 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 numbertheoretic 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[edit]
I don't think that the section on "Jaina mathematics" should remain in the article.
 I don't think it's mathematical so much as philosophical. The different classes don't seem to correspond to actual infinite numbers.
 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 . Spacefilling 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.
 The section is unreferenced.
 I wasn't able to reference the section. sacredtexts.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.
 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.
 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[edit]
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"[edit]
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"[edit]
 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 Lfunctions—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[edit]
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" ??? Puhleeze.Daqu (talk) 08:11, 1 April 2009 (UTC)
Number theory as pure math[edit]
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 outnumbered. 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 outnumbering yours is sufficient to decide the content of this encyclopedia. RobHar (talk) 16:05, 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[edit]
(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)
Misuse of sources[edit]
Jagged 85 (talk · contribs) 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?[edit]
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 restarting this article from scratch. Gandalf61 (talk) 17:11, 4 July 2010 (UTC)
 Actually, I think this article is rather bad. So bad I've mostly never dared improving it. I think little justification is required to restart from scratch. I have no idea why this article is rated a B. I would recommend the rewrite be done in a sandbox first, though (at least to get it to some level). RobHar (talk) 22:07, 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 nonnumbertheorist, 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 nonnumbertheorists 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 writeup. RobHar (talk) 14:09, 20 September 2011 (UTC)
"The Higher Arithmetic"[edit]
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. — Fly by Night (talk) 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 wellrespected textbook by an extremely wellrespected 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[edit]
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 "floatingpoint 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"[edit]
Speaking as a numbertheorist 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 IPeditor 
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 HardyWright, and much of the qseries 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 nonanalytic) proofs of analytic statements.)
 I don't know whether exactly how many number theory grad students recognise qseries 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 HardyWright 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 qseries (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)
Classical vs. recent[edit]
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 wellestablished 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[edit]
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 (genericlooking projection onto 2 dimensions of) a genericlooking projection onto 3 dimensions of a fourdimensional (over the reals) plot of 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 45 which you can include in the page. Igorpak (talk) 15:56, 12 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)
Pictures redux[edit]
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?[edit]
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):
 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.
 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".
 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.
(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)
Reformat of refs and citations[edit]
In response to Garald's above question "What is to be done?", I added a bibliography section, and reformatted 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)
Diophantine approximation and Transcendence theory[edit]
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 wellwritten 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):
 * Waldschmidt, Michel (2008). "Introduction to Diophantine methods: irrationality and transcendence" (PDF). Paris: Université Pierre et Marie Curie.
 * e.g. {{sfnWaldschmidt2008pp=7178}} HTH, >MinorProphet (talk) 21:28, 30 July 2012 (UTC)
Subfields[edit]
If we are to request to be rated and be confident to be raised to Aclass  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 getthecitation.
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[edit]
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)
Unreadable sentence[edit]
In section 2.3, the last sentence of the first paragraph, is not readable to me:
 If is barely larger than , must 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[edit]
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)
This article is somewhat lacking...[edit]
Asymmetric encryption is the most significant and wellknown application of number theory (with the application of finite field arithmetic to coding theory a close second). I am astonished AE is barely mentioned by this article; the words "public key" and "asymmetric" are not even uttered, or so CtrlF tells me. AE is a highly active field of scientific research involving hundreds if not thousands of mathematicians and computer scientists and is essential to modern information security. Surely it deserves a paragraph or two? And why is Fermat lavished with attention whilst Gauss  who practically invented modern number theory  gets a meagre paragraph?
A quick list: Number theory is often called "higher arithmetic", this should go in the lede. "Arithmetical" is often opposed to "algebraic", "analytic" or "geometric" to indicate the type of mathematics being used. Additive number theory is not entirely subsumed by combinatorial number theory; for example, the circle method is an analytic, not combinatorial, tool. The padic numbers are about as number theoretic as you can get; a single link is far from sufficient. There needs to much more group theoretic material. Number theory and the properties of the integers and primes are critical to the theory of finite abelian groups; ℤ and its quotients are essential to group (co)homology. And on that note, the few remarks on Galois theory should be expanded and inserted into the text proper, not hidden in footnotes. There is sore lack of advanced number theory, such as elliptic curves, modular forms, Galois cohomology, class field theory, adeles/ideles, arithmetic topology, special functions, etc.
Unless anyone objects I will work on this article to fix these problems. Advice and (educated) opinion is invited. Asmallgreycat (talk) 02:05, 28 April 2014 (UTC)
 Please go ahead. On 20th [century] number theory  be careful to keep the material relatively accessible, and also take advantage of the opportunity to add plenty of footnotes (something that general editors may find lacking in that part of the article). I agree Gauss needs more space, though arguably not all of Disquisitiones is number theory in the modern sense of the word. Garald (talk) 22:52, 10 May 2014 (UTC)
On the history section: (a) we should take a relatively recent article Plimpton 322 (Britton, John P., Proust, Christine, and Shnider, Steve, Plimpton 322: a review and a different perspective, Arch. Hist. Exact Sci. 65 (2011), no. 5, 519–566.) into account; (b) Let us try to agree on what else to say about Gauss. The issue is that he used to be described as the originator of modern number theory, but part of Disquistiones, while extremely important, are really closer to some sort of protoGaloistheory than to number theory, and other parts simply fix notation. (What is certainly arithmetical is his work on quadratic reciprocity and on binary quadratic forms  oh, and then there is his empirical work on the prime number theory.)
On the "main" section: I agree this needs more detail and more sources. It's actually quite frustrating for a number theorist to work on this, as there are many things one wants to say which one cannot say until one finds a source in the literature saying as much. Garald (talk) 13:23, 26 September 2014 (UTC)
Algorithmic undecidability[edit]
There's a dispute about the following text:
 On a different note — some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven that there is no Turing machine which can solve all Diophantine equations (see Hilbert's 10th problem).
and
 There are thus some problems in number theory that will never be solved. We even know the shape of some of them, viz., Diophantine equations in nine variables; we simply do not know, and cannot know, which coefficients give us equations for which the following two statements are both true: there are no solutions, and we shall never know that there are no solutions.
The second assertion does not follow from the first. Just because there is no Turing machine that can determine for every Diophantine equation whether or not it has a solution, it does not follow directly that there is a Diophantine equation for which we cannot know whether or not it has a solution. It only means that we cannot know it simply by feeding it into a Turing machine. It does not preclude our knowing it some other way. A reference to a reliable source would be helpful here, such as Davis, Martin; Matiyasevich, Yuri; Robinson, Julia (1976). "Hilbert's Tenth Problem: Diophantine Equations: Positive Aspects of a Negative Solution". In Felix E. Browder. Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. XXVIII.2. American Mathematical Society. pp. 323–378. ISBN 0821814281. Reprinted in The Collected Works of Julia Robinson, Solomon Feferman, editor, pp.269–378, American Mathematical Society 1996. In this, on page 326, we read it may be possible to devise ad hoc methods for particular equations. Deltahedron (talk) 10:59, 27 September 2014 (UTC)
 Surely a formal mathematical proof is also something that can be verified by a Turing machine? (What other method would there be  divination?) We can run a Turing machine on the set of all possible formal proofs, verifying which ones are valid, and then test whether an equation it is given satisfies the conditions of the statements (provided that the conditions, too, can be stated formally). What the statement in page 326 presumably means is that there can be methods for particular families of equations; indeed, this is what a large part of Diophantine analysis is about. Garald (talk) 11:08, 27 September 2014 (UTC)
 The point is that the text we write is not what you, I or anyone else thinks might be correct: we report what independent reliable sources say. If one assertion follows from another so transparently that it constitutes a routine calculation, that is not original research, of course. In this case there is a distinction between proving statements within the system and proving statements about the system which can only be written about using wellchosen references. It is certainly not a trivial deduction. If the assertion that there is a Diophantine equation for which we can never know whether or not it has a solution is correct, then it will be easy to find a reference that says so, and the burden is on the editor who wants to include it to find such a reference. Deltahedron (talk) 11:24, 27 September 2014 (UTC)
 The problem here is something that is frustrating overall: (a) what is routine for some is not routine for others, (b) no doubt we could find references (in the semipopular, allegedly reliable literature) that supports many misleading statements. These are not facts in the physical world (for which the reliable sources/verifiability policy was meant), but statements in a topic in which some things follow from others.
 That said  whether a Turing machine raises a little flag on being fed a diophantine equation, or whether a number theorist proves a statement about solutions to diophantine equations (giving an actual formal proof, let us say), aren't we working at the same level? Neither is a statement about the system in the sense that Hilbert's Tenth Problem is. Garald (talk) 11:34, 27 September 2014 (UTC)
 The word "know" is not synonymous with "prove in a given formal system". For example, Gödel's incompleteness theorems considers statement in a formal system that are true but not provable in that system. However, this is slightly beside the point. As I stated before, if there is a reliable source that supports the text, by all means cite it. If there is no such source, we cannot use the text. Wikipedia:Verifiability policy applies to articles on every subject. Deltahedron (talk) 11:46, 27 September 2014 (UTC)
 That is the difference between "truth" and "provability", not a difference between "knowledge" and "provability". I would agree that "we will never be able to prove" would be better than "we will never know", as this is not the place to discuss the meaning of "knowledge". At any rate, the important issue here is not this particular passage, but that of how we are ever supposed to write a general mathematical article. Yes, we should use referenced statements and their logical consequences. Of course, in the process of explaining a statement to a general audience, we are using our knowledge of the meaning of technical terms, as well as a measure of experience and technical common sense. How would we answer to claims that such work (which inevitably contains a small interpretative component) constitutes original research, simply because the layperson does not see the technical statement in the literature and a statement made here as obviously equivalent? (If we are writing an article at all, it is precisely because technical statements would not always be understandable in their original form.) Garald (talk) 12:00, 27 September 2014 (UTC)
 I think there are two main points:
 1. The question "are there problems that will never be solved?" is contentious. Garald believes the answer is yes, others (e.g. Kurt Gödel, Deltahedron and I) believe the answer is no. Sources:
 Whether there exist socalled "absolutely undecidable" statements, whose truth value can never be known or is illspecified, is a controversial point among various philosophical schools. Undecidable problem
 Gödel was inclined to deny the possibility of absolutely unsolvable problems [4]
 2. The rules of the English Wikipedia are pretty clear, I think:
 ...any material whose verifiability has been challenged ..., must include an inline citation that directly supports the material. Any material that needs a source but does not have one may be removed. Wikipedia:Verifiability
 What can we do? I think we should either remove the contentious section, or rewrite it such that it reflects the fact that the philosophical implications of undecidability are controversial. Chrisahn (talk) 13:29, 27 September 2014 (UTC)
 Also, before saying that a problem will never be solved, one has to provide a definition of what means "solving a problem". IMO, the proof of nonexistence of an algorithm for solving every Diophantine equation is a solution of Hilbert's tenth problem. In other words the proof of nonexistence of some type of solution is itself a solution. We must accept this as a solution. Otherwise, Cube duplication would an example of a problem that will never be solved. Moreover, the undecidability of a problem with a given set of axioms is different from the possibility of proving that it has no solution. For example, it is not clear for me if Wiles' proof of Fermat's Last Theorem may be expressed with only Peano axioms. If not, Fermat's last theorem could be undecidable in Peano arithmetic. In other words, saying that some problems will never been solved is unverifiable speculation; therefore, not only a source is needed, but also this assertion must be presented as the opinion of the author of the source. D.Lazard (talk) 15:15, 27 September 2014 (UTC)
 Just to supplement Chrisahn's comment above: I don't want to advance any particular position about whether there exist absolutely insoluble problems; what I do say is that such a claim does not follow as an immediate logical consequence of the nonexistence of a Turing machine that resolves all Diophantine equations. It's not that I fail to see the connection because I'm a "layperson": it's because it is actually not an immediate logical consequence. Deltahedron (talk) 15:49, 27 September 2014 (UTC)
 Look, this is a problem we can get around by careful wording, without getting into contentious philosophical issues. Let us also try not establish an interpretation of Wikipedia:Verifiability that would make it impossible to write a mathematics article. Let us stay away from the term "knowledge" and focus on "proof", spacifying that we are working within a given set of axioms. And yes, of course a Turing machine can check formal proofs. My only doubt here is what we can allow ourselves to say about what the set of axioms should be; we should say "a recursive set of axioms", but that may be too technical, though again we can provide a link. We could say: "... there is no Turing machine that can solve all Diophantine equations. In particular, this means that, given a recursive set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (Necessarily, we would be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove, of course, that a particular Diophantine equation is of this kind, since this would imply that it has no solutions, and that is precisely what we cannot prove.)" I suppose anybody can figure out the thing between parenthesis; I'm thinking we should add it simply because it answers a common question. We could change "a recursive set of axioms" by "any standard set of axioms", which should be clearer to the layperson, but at the same time "standard set of axioms" has no precise technical meaning; I really have no position about that. Garald (talk) 22:15, 29 September 2014 (UTC)
 The sources quoted here should be useful: [[5]] Garald (talk) 16:27, 30 September 2014 (UTC)
 Looks good! Suggestion: Replace recursive (which unhelpfully redirects to recursion) by recursively enumerable. It's more precise and hopefully less confusing for laypeople because the word enumerable is easier to grasp than the word recursive. Chrisahn (talk) 23:41, 30 September 2014 (UTC) P.S. I guess in some rather exotic axiom systems / inference rules, a statement of a solution of the equation may be a theorem while the statement asserting the existence of a solution isn't... ;)
 computably enumerable may be even better. Chrisahn (talk) 12:16, 2 October 2014 (UTC)
 Just to supplement Chrisahn's comment above: I don't want to advance any particular position about whether there exist absolutely insoluble problems; what I do say is that such a claim does not follow as an immediate logical consequence of the nonexistence of a Turing machine that resolves all Diophantine equations. It's not that I fail to see the connection because I'm a "layperson": it's because it is actually not an immediate logical consequence. Deltahedron (talk) 15:49, 27 September 2014 (UTC)
 Also, before saying that a problem will never be solved, one has to provide a definition of what means "solving a problem". IMO, the proof of nonexistence of an algorithm for solving every Diophantine equation is a solution of Hilbert's tenth problem. In other words the proof of nonexistence of some type of solution is itself a solution. We must accept this as a solution. Otherwise, Cube duplication would an example of a problem that will never be solved. Moreover, the undecidability of a problem with a given set of axioms is different from the possibility of proving that it has no solution. For example, it is not clear for me if Wiles' proof of Fermat's Last Theorem may be expressed with only Peano axioms. If not, Fermat's last theorem could be undecidable in Peano arithmetic. In other words, saying that some problems will never been solved is unverifiable speculation; therefore, not only a source is needed, but also this assertion must be presented as the opinion of the author of the source. D.Lazard (talk) 15:15, 27 September 2014 (UTC)
 What can we do? I think we should either remove the contentious section, or rewrite it such that it reflects the fact that the philosophical implications of undecidability are controversial. Chrisahn (talk) 13:29, 27 September 2014 (UTC)
Citation #1 not understood[edit]
The first citation to this article ([1]) is not clear. Can it be made a bit understandable? It seems like it refers to a page #1 of a book by Long, published in 1972, but it requires further clarification. Also, I found that, the following two links refer to the same fact: [2], [3].
I am sorry, if I am making a small issue big here! Anubhab91 (talk) 14:29, 8 October 2014 (UTC)
[1] http://en.wikipedia.org/wiki/Number_theory#CITEREFLong1972
[2] http://www.gresham.ac.uk/lecturesandevents/thequeenofmathematics
[3] http://www.storyofmathematics.com/19th_gauss.html
Integers, or natural numbers?[edit]
Most sources describe number theory as the study of positive integers, or natural numbers. I would change that. 206.116.67.167 (talk) 01:27, 7 July 2015 (UTC)
 Can you provide a reliable source supporting that "most sources ..."? In any case, the study of integers includes the study of subsets of integers, including positive integers. On the other hand, one of the first important properties of natural numbers is that they are naturally embedded in an Abelian group called the integers. So, the study of the natural numbers is exactly the same as the study of integers. D.Lazard (talk) 06:57, 7 July 2015 (UTC)
 I was about to raise exactly that same point. Here are a couple sources googled for online:
 http://www.math.brown.edu/~jhs/frintch1ch6.pdf: "Number theory is the study of the set of positive whole numbers 1, 2, 3, 4, 5, 6, 7, . . . , which are often called the set of natural numbers..."
 http://www.britannica.com/topic/numbertheory "Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). "
 I was about to raise exactly that same point. Here are a couple sources googled for online:
 I note that http://mathworld.wolfram.com/NumberTheory.html MathWorld uses the term "whole numbers" which can be considered a bit sloppy and ambiguous.
 I also note that http://www.bristol.ac.uk/maths/people/group/mathsthemes/5026 specifically says integers: "Commonly referred to as the queen of mathematics, number theory is an ancient branch of pure mathematics that deals with properties of the integers."
 Might it be worth mentioning in the preamble that some sources say one thing and some another? That is the approach taken by the generally utterly appalling ProofWiki:
 http://proofwiki.org/wiki/Definition:Number_Theory: "Some sources allow that number theory studies the properties of all integers, not just the natural numbers, that is, the positive integers."
 It's a minor point, but from the point of view of WikiPedia being an encyclopedia, it's worth being encyclopedic about it. Matt Westwood 08:09, 25 October 2015 (UTC)
 I have edited the article for asserting the number theory is "the study of the natural numbers and the integers". This allows to avoid such a unnecessary debate. In fact, for every scientific area, not only number theory, one may have a useless debate to define the limit of the area. In this case, as studying integers and studying natural numbers is exactly the same thing, the debate is even more useless. Also it should be pointed that, if some sources say one thing and some source say another thing, no source say that number theory is not one of the things. Thus, IMO, it is unnecessary to "mention in the preamble that some sources say one thing and some another". D.Lazard (talk) 09:16, 25 October 2015 (UTC)
 I disagree with your statement that "studying integers and studying natural numbers is exactly the same thing", but I agree with pretty much everything else you're saying. And I like your changes to the article. That's all. Since Wikipedia talk pages are usually full of discord, I just wanted to say something nice for a change. Have a nice day everyone. :) Chrisahn (talk) 22:27, 27 October 2015 (UTC)
 I have edited the article for asserting the number theory is "the study of the natural numbers and the integers". This allows to avoid such a unnecessary debate. In fact, for every scientific area, not only number theory, one may have a useless debate to define the limit of the area. In this case, as studying integers and studying natural numbers is exactly the same thing, the debate is even more useless. Also it should be pointed that, if some sources say one thing and some source say another thing, no source say that number theory is not one of the things. Thus, IMO, it is unnecessary to "mention in the preamble that some sources say one thing and some another". D.Lazard (talk) 09:16, 25 October 2015 (UTC)
 It's a minor point, but from the point of view of WikiPedia being an encyclopedia, it's worth being encyclopedic about it. Matt Westwood 08:09, 25 October 2015 (UTC)
As D. Lazard  it boils down to the same thing. We (in math) don't say "natural numbers" all that often nowadays, since nobody can agree on whether they include 0. So, "the study of the integers" would be best. Garald (talk) 14:10, 15 January 2016 (UTC)
Assessment comment[edit]
The comment(s) below were originally left at Talk:Number theory/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
Needs more information about modern advances in number theory and a more comprehensive timeline of significant milestones in number theory. Also has some awkward spots. shotwell 06:18, 6 October 2006 (UTC) 
Last edited at 23:04, 19 April 2007 (UTC). Substituted at 01:36, 30 April 2016 (UTC)
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