As some of you are aware, I’ve done some work on a possible replacement for Wikipedia’s Number Theory page (which is arguably fairly dreadful at the moment, for a variety of reasons). I think the time has come to put the new version up. For the moment, it’s at this temporary home, side by side with my article on the history of number theory.

I’ve been meaning to put them on Monday. Feel free to comment either here or on the corresponding talk pages!

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## About valuevar

I am a number theorist with side interests in combinatorics and group theory.

Some suggestions:

Use more links! For instance, the first two sentences should, in my opinion, have links to the wikipedia entries on “integers”, “rational numbers”, “mathematics”, “algebraic integers” and prime numbers. In general, the article should, throughout, have MANY more links.

The current wikipedia entry says that number theory is a branch of pure mathematics, and you have replaced it simply by mathematics. I can see why one would prefer to do this, but I prefer “pure mathematics” because a) if pure mathematics should mean anything, it should include number theory b) this is closer to how I and most number theorists I know feel about their subject.

I think you should retain some of the old history of the current wikipedia entry.

Sentences like this: “Some of the main questions are those of distribution: questions, say, on patterns or their absence (in the primes or other sequences) or, more generally, questions on size, number and growth” feel simultanoeusly too vague (to me, speaking as a number theorist) and too technical (when put on my layperson hat and imagine a general reader looking up wikipedia). In general I think such sentences are not for the introduction, and for this reason, I actually prefer the curent introduction to your suggested replacement.

However, I see the reason why you want some of these changes, and if I were to suggest a new introduction that includes many of your elements, it would be something like:

*****

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study integers either in themselves or as solutions to equations (diophantine geometry). They also study the prime numbers (the building blocks of integers) as well as various objects that are made out of integers (such as rational numbers) or defined as generalisations of the integers (such as algebraic integers). The questions in number theory are often best understood through the study of analytical objects that encode number-theoretic objects in some fashion (analytical number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).

The older term for number theory is arithmetic; it was superseded by “number theory” in the nineteenth century, though the adjective arithmetical is still fully current. By 1921, T. L. Heath had to explain: “By arithmetic Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers.”[1] The general public now uses arithmetic to mean elementary calculations, whereas mathematicians use arithmetic as this article shall, viz., as an older synonym for number theory. (The use of the term arithmetic for number theory has regained some ground since Heath’s time, arguably in part due to French influence.[2] In particular, arithmetic is preferred as an adjective to number-theoretic. Moreover, “the arithmetic of” is used, whereas “the number theory of” is not; thus, for example, the arithmetic of elliptic curves.)”

Hi –

Thanks for the comments!

* I agree the introduction had to be made spiffier. Thanks for your help.

* With all due respect for whomever, the history section in the current version is simply bad. Things that are not germane to the main issues are introduced, there are anachronistic statements all over the place, etc.

I wrote a much longer history section, and got criticised for its length. I’ve moved it to its own article (“History of number theory”) to which a link is provided. In the new version of the main page, the history section is reduced to a summary. I think this might be enough – the main purpose of the history section in the general article is to provide an introductory narrative. Please tell me if you disagree.

(Also, it’s best to move this to the talk page at wikipedia now.)

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En español tambien es muy poco lo que mencionan.

Tambien falta tu biografia, no tienes biografia en español y es tu lengua madre