Talk:Number theory

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Mathematics rating: B Class Top Priority  Field: Number theory
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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)

Contents

[edit] 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 (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)

[edit] 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)

[edit] 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).

[edit] 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)

[edit] 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)

[edit] how about

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)

[edit] Please explain the need for this page

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)

[edit] 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 \aleph0. 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)

[edit] 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)

[edit] "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)