Talk:Number theory

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

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

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

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

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

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

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

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

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

Ooops.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Difficulty Level

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

For instance, I think the average university educated person has trouble grasping the concept of algebraic numbers, probably never having heard the words together before. As such, it seems to me best just to mention algebraic numbers as being a type of number generalizing the integers as in the introduction, but to keep the discussion of algebraic number theory brief. Certainly, topics in the last two paragraphs or the section about algebraic number theory, namely number fields, field extensions, Galois theory, class field theory, and the Langland's program (!!) are waaaaaaaay too sophisticated even to mention to the average university educated person. This material would be completely lost on them. These topics get so advanced that I would find it hard to argue that class field theory and the Langland's program should even be included on the "advanced" page. Why not include these in a page on algebraic number theory, or even something more advanced than that?

I think we can assume that the "typical university educated person" that we are supposed to aim at will have some exposure to calculus of one variable. In the analytic number theory section, it is noted that "analytic methods" refers to calculus-like methods, which is good. But perhaps we can mention that "analytic" refers to calculus-like methods when the word "analytical" is used in the introduction? (Which reminds me, I wish I knew exactly when adding an "al" to words ending in "ic" is appropriate -- geographic/geographical, geometric/geometrical often seem used synonymously as adjectives. Arithmetic and logic are different, because they function as nouns as well. I tend to favor "analytic" rather than "analytical", because they seem synonymous to me and I favor brevity. Any opinions on this?)

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

The section on Diophantine Geometry talks about n-dimensional space, counting the holes in a surface in 4-dimensional space (!!), and genus. I don't think the casual reader will get anything out of this discussion. Why not give a simple example -- say x^2 + y^2 = z^2, then mention Fermat's last theorem. Then say diophantine geometry is the study of similar types of problems with more general polynomial equations. The rest of this stuff should be placed on a MUCH more advanced page (in my opinion, it is too advanced even for the main diophantine geometry page).

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

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