Talk:Number theory

Citation #1 not understood
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].

[1] https://ingen.miraheze.org/wiki/Number_theory#CITEREFLong1972

[2] http://www.gresham.ac.uk/lectures-and-events/the-queen-of-mathematics

[3] http://www.storyofmathematics.com/19th_gauss.html

I am sorry, if I am making a small issue big here! Anubhab91 (talk) 14:29, 8 October 2014 (UTC)

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 Ingenpedia 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
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)

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

The page will be discussed at Ingenpedia: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)

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)