Talk:Number theory

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Integers, or natural numbers?

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/number-theory "Number theory, branch of mathematics concerned with properties of the positive integers (1, 2, 3, …). "

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/maths-themes/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:

https://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)

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

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)

Gauss, Fermat

The editor who commented further up that Fermat got a bit too much space and Gauss too little has a point. (That emphasis is, in retrospect, an indirect reflection of that in Weil's book.) I've pared down Fermat's section slightly, without removing anything essential. Gauss needs his own section. Who wants to write it? Garald (talk) 11:44, 26 February 2019 (UTC)

GA Review

This review is transcluded from Talk:Number theory/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: David Eppstein (talk · contribs) 01:23, 3 March 2019 (UTC)

I note that nominator Justlookingforthemoment appears never to have edited this article, This is not a rule violation, but it is a bit of a red flag for caution that improvements needed to reach GA status may not be forthcoming. Additionally, it means the nominator probably did not conduct an informal review for what improvements might be needed to reach GA and then perform those improvements. In particular, the article has a valid "Section needs expansion" tag dating from March 2016.

This is a very broad topic, one that I would expect an article to hit the highlights of and leave the details of for other articles. Roughly half the article is devoted to the history of number theory and another half to the subfields of number theory and their major results. This is reasonable, but I don't understand why half of the subfields are grouped into "Main subdivisions" and half into "Recent approaches and subfields". There is also a stub of an Applications section but this is inadequate and tagged as inadequate.

The history section is generally well sourced but the remaining sections are not. The references appear to be generally consistently formatted but Sachau 1888 is missing, the undated Apostol citation has an error message because of the use of a template in its series parameter, and Milne 2014 uses a bare url, not properly formatted with the template.

The images appear to be relevant and appropriately licensed but the Plimpton 322 image is associated with article text that takes only one specific and contentious interpretation of the meaning of that tablet.

Because "It has, or needs, cleanup banners that are unquestionably still valid", it meets the quick fail criteria in WP:GAFAIL.

—David Eppstein (talk) 02:22, 3 March 2019 (UTC)

It was a surprise to me that someone nominated the page now. It is clear that there were some minor issues that absolutely had to be addressed first (the section that needs expansion, the errors in citations) and some broader issues that needed more work (sources for the second half of the article -- though that's a bit of a tough issue, since many of the statements are second nature to a professional, and hence hard to source). Still, it is very helpful to have feedback.

Two issues:

• What do you mean precisely by "one specific and contentious interpretation of the meaning of the tablet"? The :tablet does contain a list of what is conventionally called Pythagorean triples, and they are labelled as such. As :for applications, the field is indeed wide open, but we mention at least two opposing views. We could also include a :more recent response to Robson - is that what you imply is missing?
• As for why "half of the subfields are grouped into "Main subdivisions" and half into "Recent approaches and :subfields"" - it is more or less clear that some subfields are much newer and well defined than others (the name :"additive combinatorics" is less than 20 years old, though the field has been around since the 1960s, or in some :sense for longer). Does the division seems too arbitrary or unnecessary? If so, we can talk about removing it, but I :am sure I am not the only one who wonders where exactly the problem lies.

Also: wouldn't nominating the article for a B-class review be a logical first step, once the issues above are addressed? Garald (talk) 03:25, 19 March 2019 (UTC)

"The references appear to be generally consistently formatted" - I would hope so - I went through every single reference in the article as it stood in April 2012 and formatted the lot with consistency uppermost in my mind. Since then, additions have been made by others who apparently just don't care. @Garald: Would you like me to have another go? I was randomly looking through my previous "achievements", fixed the Milne cite and then wondered how the article was doing, hence this... PS I used to think there was only one infinity until I came across Aleph-null in around 2007. PPS Still hung up on that old cow problem...— Preceding unsigned comment added by MinorProphet (talk • contribs) 20:20, 7 April 2020 (UTC)

Nomination of Portal:Number theory for deletion

A discussion is taking place as to whether Portal:Number theory is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The page will be discussed at Wikipedia:Miscellany for deletion/Portal:Number theory until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the page during the discussion, including to improve the page to address concerns raised in the discussion. However, do not remove the deletion notice from the top of the page. North America¹⁰⁰⁰ 21:10, 26 May 2019 (UTC)

Image in lead paragraph (Ulam spiral)

The image in the lead paragraph has been replaced by one of the Ulam spiral. The legend reads "The prime factorisation of the integers is a central point of study in number theory and can be visualised with this Ulam spiral variant. Number theory seeks to understand the properties of integer systems like this, in spite of their apparent complexity." Unfortunately, several things here seem to be a ltitle off. An Ulam spiral depicts primality, not factorization. It's unclear what is meant by "integer system" here, or even "apparent complexity".

The Ulam spiral may not be a very good choice for an image in the lead: it gives the reader the illusion of some imperfect patterns (due to small-number effects), whereas it is a standard conjecture that the statistical tendency towards such patterns is zero, once trivial local effects are set aside. Garald (talk) 16:04, 8 August 2019 (UTC)

PS. Some of those initial patterns do have number-theoretical significance, but discussing that involves algebraic number theory and would probably take us too far afield. Garald (talk) 11:59, 30 January 2021 (UTC)

Pythagorean triples

The beginning of "The Dawn of Arithmetic" has been edited (almost certainly by a non-number theorist). It now reads: "The world's oldest document about Mathematics is the Berlin Papyrus 6619 from the Middle Kingdom,[2] second half of the 12th (c. 1990–1800 BC) or 13th dynasty (c. 1800BC–1649BC),[3] and had a problem similar to the Pythagorean theorem before Pythagoras lived and much before Euclid (300BC). Another early historical find of an arithmetical nature..."

This is off-topic; the Pythagorean theorem is not in itself a statement about number theory, or "of arithmetical nature". This information on the Berlin Papyrus belongs in a footnote (and the implication that it is "of arithmetical nature" should be avoided). A table of rational Pythagorean triples is another matter altogether. Garald (talk) 16:15, 8 August 2019 (UTC)

Incidentally: is "not to be confused with Numerology" really necessary? It's the equivalent of having a four-letter word in the first sentence. Garald (talk) 11:11, 14 October 2019 (UTC)

Specialists, please edit

Something is striking - edits (minor and not always good) seem to come largely from amateurs or at least non-specialists; while some number theorists do edit the talk page, barely any edit the page itself. Specialists: be bold and edit. Garald (talk) 12:58, 14 October 2019 (UTC)

Combinatorial number theory

The number theory navbox has a link to "Combinatorial number theory" but that just links to a non-existent section on the number theory article. Should the link be removed or should the number theory article have a section for "Combinatorial number theory"? I notice there is a section called "arithmetic combinatorics" - is that just a modern name for "Combinatorial number theory"? Fdfexoex (talk) 03:07, 21 November 2019 (UTC)

In the Number theory navbox, I have replaced Combinatorial number theory with Arithmetic combinatorics. Combinatorial number theory was a link to a section of Number theory that was deleted 02:50, 10 October 2011. Arithmetic combinatorics is the closest replacement. Thank you for pointing this out. — Anita5192 (talk) 04:51, 21 November 2019 (UTC)

Takiltum

(related to reference 2 on the term takiltum being problematic - btw, one would expect to be able to click on the term takiltum to see some article on what it means) "But the author of Plimpton 322 did not have a modern viewpoint. According to Robson, the p/q theory fails to account for many of the features of the tablet, including that fact that it records values of (c/a)2 instead of a. The reciprocal pair explanation, she says, makes more sense in light of what’s been learned about Old Babylonian tablets in the last half century. One key is the label for the first column. Neugebauer and Sachs rendered it as “The takiltum of the diagonal which has been subtracted such that the width...,” leaving takiltum untranslated and the label unfinished, because part of it near the end is unreadable. (“Diagonal” means “hypotenuse,” since right triangles arise by cutting a rectangle diagonally in half. “Width” and “short side” are also synonymous.) Subsequent scholars, observing the use of takiltum in other mathematical tablets, determined that it refers to a “helping” or “holding” number. With that meaning and an educated guess for what makes grammatical sense (and also fits physically) in the unreadable and damaged portions, Robson offers a new translation: “The holding-square of the diagonal from which 1 is torn out, so that the short side comes up.” That reading, she says, aligns well with the Old Babylonian approach to solving reciprocal-pair-type problems and with other mathematical tablets of the time. So it seems that the author of Plimpton 322 was no lone genius—but he was probably a very good teacher." https://www.ams.org/publicoutreach/happ5-history.pdf — Preceding unsigned comment added by 46.246.247.51 (talk) 04:14, 17 February 2020 (UTC)

adding a picture of sir andrew

is it fair to include a portrait of sir andrew wiles on the Number Theory page, considering there is a picture of erdos and terry?

i mean, he did prove The Last Theorem, right?

it seems there is a bit of text dedicated to this theorem on the page, right?

from what i can see, it's mentioned under Early modern number theory, subsections fermat, euler, and 'lagrange, legendre and gauss'.

i know some would say he's not any of the people on this page, the 'lowest' probably being Erdos or terry. i respect all these guys.

but my view is if Erdos gets a spot, so does sir andrew. he did prove it. and he deserves some recognition outside of the fermat's last theorem page.

just my 2c

before i forget, just for redrose74: "198.53.108.48 (talk) 23:10, 23 June 2021 (UTC)"

Adding his pictures seems okay, given the other portraits. --A D Monroe III^(talk) 14:37, 25 June 2021 (UTC)