Talk:Erdős–Straus conjecture/GA1

GA Review

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Reviewer: HenryCrun15 (talk · contribs) 04:27, 8 January 2022 (UTC)

Volunteering

Hi, I will review this article. HenryCrun15 (talk) 04:27, 8 January 2022 (UTC)

Review against the GA criteria

Thank you for the opportunity to review this article. I found the subject fascinating. A couple of points before I continue with my review:

• My apologies for rushing in and incorrectly editing the article when I first started working on this review. I had clearly skimmed the article and jumped to certain conclusions. Thank you for your corrections
• I have a Bachelor's degree in mathematics, but no further, so my ability to comment on highly specialised concepts in number theory will be limited.
• I got a Good Article mentor to look over my draft review. Thanks to Nikkimaria (talk · contribs) for the advice, and I have incorporated their suggested changes.

My comments on the article are below:

Rate	Attribute	Review Comment
1. Well written:
	1a. the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct.	Overall: The criterion of "understandable to an appropriately broad audience" is important here. The conjecture itself is something that can be understood by an undergraduate or even high school maths student, so the article should as much as possible be clear to such people. The article often uses advanced number theory concepts that only an expert in number theory would recognise, let alone understand. This doesn't mean that the advanced mathematics should be removed from the article – far from it – but it is valuable to structure the article with the considering the principles of "put the least obscure parts of the article up front" and "write one level down". With this in mind, many of my comments relate to keeping the simpler parts simple and the complex parts separate. Formulation: Move the discussion of the "distinct" variation out of this section, to keep the definition as clear as possible. Move the alternate formulation of the conjecture, $4xyz=n(xy+xz+yz)$, from its current section into the Formulation section. Background: Move the explanation of "any set of unit fractions can be converted to a set of distinct unit fractions" out of this section. The paragraph beginning The greedy algorithm for Egyptian fractions, first described in 1202... appears to be explaining two separate things: that the number of unit fractions needed to express 2/n and 3/n is well known, so 4/n is the smallest number like this that is unknown; and, it is known that every 4/n needs at most four terms. Consider separating these concepts into distinct paragraphs. Modular identities: I recommend replacing this section with a section called "Approaches to the conjecture" or similar, with subsection covering each approach. The Hasse principle for Diophantine equations asserts that an integer solution of a Diophantine equation should be formed by... What is meant by "should" here? Is it that there is a solution using the following method, but it has not yet been determined? Does it mean that researchers should use this line of inquiry? Or something else? Expand the section on computational verification. Currently, the idea that searches for counterexamples can ignore composite numbers (and why) only appears in the lead section. Usually, the lead only repeats information from the rest of the article. Similarly, only the lead contains the idea that if a counterexample to the Erdős–Straus conjecture exists, the smallest n forming a counterexample must be a prime number congruent to one of six values modulo 840, so this should also appear in the discussion on computational verification. Negative-number solutions and Generalizations: I recommend having a single section on variations to the Erdős–Straus conjecture. This article currently covers three: a restriction where the unit fractions must be distinct; a loosening where negative unit fractions are allowed; and the general problem where 4/n is replaced with k/n. The proof that the conjecture is true when negative unit fractions are allowed seems incomplete. It shows (two ways) that odd values of n have solutions, but doesn't show how even values have solutions. If so, add this in. Elsholtz & Tao (2013) have shown that... – replace "have shown" with "showed". ...if negative values were allowed the problem could be solved trivially via one of the two identities - Remove the word "trivially" and add a colon at the end.
	1b. it complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation.	Lead section: The detail that computer searches only need to look at prime number congruent to one of six values modulo 840 seems too detailed for the lead section. Merge the fourth and fifth paragraphs since they both cover the concept of computer searches for counterexamples. If you ensure that the body of the article contains all the material touched on by the lead section, then you shouldn't need inline citations in the lead itself. As discussed in MOS:LEADCITE, the lead section should generally cover material that is also covered in the article (and sourced there) and inline citations in the lead should be typically used only for "complex, current, or controversial subjects". Remove from the lead but proving it for all n remains an open problem as this concept appears twice already. Maybe link the phrase "unproven statement" to the open problem article in the first sentence. The article is good for all other style guidelines.
2. Verifiable with no original research:
	2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.	There doesn't seem a need for a link to the Tao blog post in the External links section given it is already linked in the notes.
	2b. all inline citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines.	All sources appear reliable and not controversial.
	2c. it contains no original research.	I note that the article does cite a paper published by the nominator and main contributor. I am fine with this because the paper was published in a journal, which presumably means it was peer-reviewed.
3. Broad in its coverage:
	3a. it addresses the main aspects of the topic.	It isn't stated in the article why the Erdős–Straus conjecture is important. What would be the implications if it was proven true? And if it were proven false? Any information on why people investigate this conjecture would be useful. Some researchers additionally require that the integers x, y, and z be distinct from each other, while others allow them to be equal. - But the next sentence says that it doesn't matter if they're required to be distinct unless n=2. Why do some researchers investigate this alternative of the conjecture? If Erdős didn't publish the conjecture until 1950, how was he able to work on the conjecture in 1948? Was he working with Erdős or Straus, or did he hear about it another way? When did Sierpiński propose the k=5 version, and when did Schinzel propose the fully generalized version?
	3b. it stays focused on the topic without going into unnecessary detail (see summary style).	Agreed, though as noted above some detail should be rearranged to keep individual sections on topic.
	4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each.	No problems here
	5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute.	No problems here
6. Illustrated, if possible, by media such as images, video, or audio:
	6a. media are tagged with their copyright statuses, and valid fair use rationales are provided for non-free content.	There is no media in this article (beyond presenting some mathematical expressions and equations). I am not aware of any media that would be suitable for this article.
	6b. media are relevant to the topic, and have suitable captions.
	7. Overall assessment.	It's my proposal to put this review on hold until the above questions and comments are been responded to.

The article has now been significantly rearranged in reponse to these comments. Some more detailed remarks:

Lead

All comments under 1b done. I also added a sentence at the end summarizing the generalizations section.

Formulation

Fine, both moved. I'm not convinced that moving the polynomial version up is an improvement, but it also doesn't really hurt.

Background

Explanation for why indistinct fractions can be made distinct moved to a new section.

Worst-case length for each numerator and behavior of the greedy algorithm separated into two paragraphs.

Modular identities

Split into two sections and combined with the theoretical part of the "number of solutions" section. The computational part of the "number of solutions" section has instead been merged into the other computation section.

Attempted to explain Hasse principle more clearly

The section on computational verification does not belong in this section, but I did repeat in more detail the explanation for why the search can be limited to primes.

The " one of six values modulo 840" is not a computational result, it is a mathematical result. It is incorrect that this statement appeared only in the lead. It is a summary of a statement later in the article, "except possibly those that are 1, 121, 169, 289, 361, or 529 mod 840".

Negative-number solutions and Generalizations

The negative-number part is not really a generalization. Or if it is it's a trivial generalization that nobody cares about because it's trivial. Really it's an explanation for why the formulation of the main problem uses positive numbers. I moved it into the formulation section. Same goes for the distinct vs not distinct part.

"The proof that the conjecture is true when negative unit fractions are allowed seems incomplete. It shows (two ways) that odd values of n have solutions, but doesn't show how even values have solutions. If so, add this in. ": This was omitted because it's so totally trivial that no source even mentions it. It's just a special case of the expansion for $n=2$ and the fact that a solution for any $n$ gives a solution for any multiple of $n$. No negative numbers needed. (Allowing them doesn't mean they are required to be used.)

Have shown => showed: Done.

Remove "trivially": done.

Add a colon at the end of the sentence.: not done.: We don't end sentences that way.: We also don't put: colons before: nouns in: sentences. The equation in this sentence is, grammatically, a noun. It doesn't need a colon.

External links

Re: "There doesn't seem a need for a link to the Tao blog post in the External links section given it is already linked in the notes.": It is false that it is already linked in the notes. The references link to a technical paper by Tao. The external link goes to a blog post of Tao, describing his work in that paper in a more accessible way (or at least trying to; I don't always find Tao's blog posts to be easy to read). They have the same title but are different links. The notes link to a different blog post by Tao with a different title.

Broadness of coverage

Re "It isn't stated in the article why the Erdős–Straus conjecture is important.": it isn't important, in the sense that (as far as I know) it has no real-world applications. Why would you think that it would? This is not the sort of topic that is studied for its importance. In any case, without sources that make editorial judgements about its importance, we cannot add those judgements to the article.

Re "Why do some researchers investigate this alternative of the conjecture?": It isn't because they want to study a different variation. It's merely that there are two different ways you can define the problem, so when you write down what problem you are defining, you are going to pick one of those two ways. It's an arbitrary choice, but it's not possible to avoid choosing.

Re "If Erdős didn't publish the conjecture until 1950, how was he able to work on the conjecture in 1948? Was he working with Erdős or Straus, or did he hear about it another way?": We can only write what our sources say. Our sources say that it was formulated in 1948 and published in 1950. You know that researchers talk to each other, both formally (in conference talks and seminars) and informally, right? Also that the journal publication process for mathematics papers can take years, with long delays sometimes occurring between coming up with an idea and polishing it to the point that it is suitable for publication, between submitting the paper and getting back a referee report, between that report and the paper's final revision and acceptance, and between acceptance and its actual appearance in the journal? So it is not surprising that Erdős and Straus talked to each other, and maybe to other people like Obláth or maybe to people who talked to other people who talked to Obláth, long before the paper made its way through the long process of getting published. A discussion of the sociology of mathematics publication in the late 1940s is probably well beyond the scope of this article.

Re when did Sierpinski and Schinzel conjecture it: They are both in Sierpinski's 1956 paper. The first part, for 5/n, is stated as "J'oserai aussi poser l'hypothèse que pour les entiers n > 1 tout nombre 5/n est un B3". Here B3 means expressible as a sum of three unit fractions. The phrasing suggests that Sierpinski is making the 5/n conjecture at that point in the paper rather than repeating a conjecture from earlier work. A few lines down he credits the generalization to his student Schinzel: "D'après une hypothèse de A. Schinzel, quel que soit le nombre naturel m donné, il n'y a qu'un nombre fini de nombres naturels n pour lesquels le nombre m/n n'est pas un B3". I added the 1956 date to the article text.

@HenryCrun15: I think now I've addressed all of the comments you made, so could you take another look, please? —David Eppstein (talk) 22:17, 16 January 2022 (UTC)

Updated review of 17 Jan 2021

Hi @David Eppstein:. I see that you've edited the article following my first review and have requested another look. My updated review is below.

Rate	Attribute	Review Comment
1. Well written:
Y	1a. the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct.	Lead section: ... certain infinite arithmetic progressions have simple formulas for their solution... Consider removing the subjective term "simple". This may be considered presumptuous language. Background and history: ... in the more modern vulgar fraction form... I see that the linked Wikipedia article on "vulgar fractions" uses the term "simple fractions" in the first instance, and throughout that article, and gives "vulgar fraction" as an alternative name. I recommend using "simple fraction" in the article on the Erdős–Straus conjecture. It appears to be more commonly used than "vulgar fraction". Further, the word "vulgar" contains negative connotations that might confuse a reader (unnecessarily, since other terms are available). Computational results: If the conjecture is false, it could be proven false simply by... Remove "simply" as it does not add anything to the sentence. Also, consider rewording this section to avoid the double use of "false", in order to improve the flow of the sentence. Overall: These three points are all minor and so I feel that this article meets this criteria.
Y	1b. it complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation.	This article complies with all these guidelines.
2. Verifiable with no original research:
Y	2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.	The blog post by Terrence Tao entitled "On the number of solutions to 4/p = 1/n_1 + 1/n_2 + 1/n_3" is a reference of this Wikipedia article, but does not appear in the article's Reference section. It needs to be added to this section so that this article "contains a list of all references (sources of information)", as required. Once added to the Reference section, the link to this blog post it should be removed from the External Links section; there is no particular reason for this article (and only this article) to be highlighted in a separate section. I am comfortable in marking this criterion as met, despite not currently technically meeting it, on the assumption that that above minor point will be carried out. While this does need to be done, it is not worth holding up the review for if it was the only issue.
Y	2b. all inline citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines.	All sources appear reliable and not controversial.
Y	2c. it contains no original research.	I note that the article cites a paper published by the nominator and main contributor. I am fine with this because the paper was published in a journal, which presumably means it was peer-reviewed.
3. Broad in its coverage:
Y	3a. it addresses the main aspects of the topic.	Negative number solutions: This section shows that the conjecture can be solved "for every odd n", but the conjecture is not limited to odd numbers. Add in a brief explanation of why even numbers meet the conjecture. Reasons why the conjecture is explored: The article does not provide information on why Erdős, Straus, Mordell, Elshotlz, etc decided to formulate and study this conjecture (that is, why did these mathematicians and others find it interesting or important). The article also does not discuss what proving or disproving the conjecture would mean for other mathematical work. I consider that a reader would be interested in both of these topics. However, the good article criteria specifically note for this criterion that: The "broad in its coverage" criterion is significantly weaker than the "comprehensiveness" required of featured articles. It allows shorter articles, articles that do not cover every major fact or detail, and overviews of large topics. Overall: I am comfortable that this article meets this criterion. As a separate note, I would observe that with a lack of sources on the reasons why the conjecture is explored, this article would likely struggle to meet the "comprehensiveness" criterion that is required for featured article status.
Y	3b. it stays focused on the topic without going into unnecessary detail (see summary style).	All good.
Y	4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each.	No problems here.
Y	5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute.	No problems here.
6. Illustrated, if possible, by media such as images, video, or audio:
Y	6a. media are tagged with their copyright statuses, and valid fair use rationales are provided for non-free content.	There is no media in this article (beyond presenting some mathematical expressions and equations). I am not aware of any media that would be suitable for this article.
	6b. media are relevant to the topic, and have suitable captions.
Y	7. Overall assessment.	I consider that all the criteria for good article status are met, and I pass this this nomination. Please do add the Terrence Tao blog post to the References section and complete the proof that, if the conjecture allowed negative unit fractions, then it would be known to be true.

Congratulations on bringing this article to Good Article status!

HenryCrun15 (talk) 05:13, 17 January 2022 (UTC)