Erdős–Straus conjecture has been listed as one of the
Mathematics good articles under the
good article criteria. If you can improve it further,
please do so. If it no longer meets these criteria, you can
reassess it. Review: January 17, 2022. ( Reviewed version). |
This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
The Swett link appears to be broken. — Preceding unsigned comment added by 63.229.7.84 ( talk • contribs) 02:41, 20 September 2005 (UTC)
I very much doubt the correctness of this section in Generalization: Hagedorn proved a related conjecture of R. H. Hardin and Neil Sloane that, for odd positive n, the equation 3/n = 1/x + 1/y + 1/z is always solvable with x, y, and z also odd and positive. A proof of this conjecture would be trivial: Let be . Ocolon 22:33, 29 January 2007 (UTC)
x, y, and z must not equal each other. I edited in a clarification. — David Eppstein 03:16, 30 January 2007 (UTC)
I prove this guess is true and if you give me any number I solve it very quickly Mohammad ghanbary ( talk) 09:06, 17 January 2021 (UTC)
A link was added to the Leibniz harmonic triangle. I wonder if this is relevant to the topic. The Leibniz harmonic triangle might be used to find solutions for but the case is rather simple anyway: Ocolon 08:28, 4 March 2007 (UTC)
The sequence A073101 requires that . In particular, this means that , and must be distinct, something which is not required by the Erdös-Straus formulation. -- Kuifware ( talk) 19:02, 28 March 2008 (UTC)
How about a couple of simple examples to illustrate? — Preceding unsigned comment added by 71.131.188.159 ( talk) 03:39, 6 July 2011 (UTC) = Like, for n=4: 4/4 = 1/2 + 1/3 + 1/6 Seems like a good example to get started with — Preceding unsigned comment added by 71.131.188.159 ( talk) 03:42, 6 July 2011 (UTC)
Since nowhere in the article does the word "distinct" appear — and in the initial statement neither "distinct" nor any synonym for it appears — the article does not make clear whether in the equation 4/n = 1/x + 1/y + 1/z the integers x, y, z are required to be distinct or not.
This ought to be obvious to anyone with any experience in writing mathematics, but: It is necessary to disambiguate anything that might readily be misunderstood, like whether x, y, z are required to be distinct.
For anyone who is certain that they know what this conjecture is: Please clarify explicitly whether or not x, y, z are required to be distinct in this article — especially right at the top of the article. Daqu ( talk) 23:50, 4 December 2014 (UTC)
In the section Negative-number solutions it says
These have only two terms on the right side, so what are x, y, and z in each case? Loraof ( talk) 16:10, 18 February 2016 (UTC)
It would be nice to give a table or list of numerical examples—each solution for, say, n up to 10 or 15. Loraof ( talk) 01:14, 8 July 2017 (UTC)
AirdishStraus ( talk) 15:23, 8 July 2017 (UTC)
فرمول این حدس را پیدا کرده ام و این حدس کاملا درست است Mohammad ghanbary ( talk) 21:45, 16 January 2021 (UTC)
I solve it Mohammad ghanbary ( talk) 22:09, 16 January 2021 (UTC)
GA toolbox |
---|
Reviewing |
Reviewer: HenryCrun15 ( talk · contribs) 04:27, 8 January 2022 (UTC)
Hi, I will review this article. HenryCrun15 ( talk) 04:27, 8 January 2022 (UTC)
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 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:
Background:
Modular identities:
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 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. |
| |
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. |
| |
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:
@ 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)
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: | ||
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. | |
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: | ||
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. | |
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 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: | ||
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:
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. | |
3b. it stays focused on the topic without going into unnecessary detail (see summary style). | All good. | |
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. | 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)
Erdős–Straus conjecture has been listed as one of the
Mathematics good articles under the
good article criteria. If you can improve it further,
please do so. If it no longer meets these criteria, you can
reassess it. Review: January 17, 2022. ( Reviewed version). |
This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
The Swett link appears to be broken. — Preceding unsigned comment added by 63.229.7.84 ( talk • contribs) 02:41, 20 September 2005 (UTC)
I very much doubt the correctness of this section in Generalization: Hagedorn proved a related conjecture of R. H. Hardin and Neil Sloane that, for odd positive n, the equation 3/n = 1/x + 1/y + 1/z is always solvable with x, y, and z also odd and positive. A proof of this conjecture would be trivial: Let be . Ocolon 22:33, 29 January 2007 (UTC)
x, y, and z must not equal each other. I edited in a clarification. — David Eppstein 03:16, 30 January 2007 (UTC)
I prove this guess is true and if you give me any number I solve it very quickly Mohammad ghanbary ( talk) 09:06, 17 January 2021 (UTC)
A link was added to the Leibniz harmonic triangle. I wonder if this is relevant to the topic. The Leibniz harmonic triangle might be used to find solutions for but the case is rather simple anyway: Ocolon 08:28, 4 March 2007 (UTC)
The sequence A073101 requires that . In particular, this means that , and must be distinct, something which is not required by the Erdös-Straus formulation. -- Kuifware ( talk) 19:02, 28 March 2008 (UTC)
How about a couple of simple examples to illustrate? — Preceding unsigned comment added by 71.131.188.159 ( talk) 03:39, 6 July 2011 (UTC) = Like, for n=4: 4/4 = 1/2 + 1/3 + 1/6 Seems like a good example to get started with — Preceding unsigned comment added by 71.131.188.159 ( talk) 03:42, 6 July 2011 (UTC)
Since nowhere in the article does the word "distinct" appear — and in the initial statement neither "distinct" nor any synonym for it appears — the article does not make clear whether in the equation 4/n = 1/x + 1/y + 1/z the integers x, y, z are required to be distinct or not.
This ought to be obvious to anyone with any experience in writing mathematics, but: It is necessary to disambiguate anything that might readily be misunderstood, like whether x, y, z are required to be distinct.
For anyone who is certain that they know what this conjecture is: Please clarify explicitly whether or not x, y, z are required to be distinct in this article — especially right at the top of the article. Daqu ( talk) 23:50, 4 December 2014 (UTC)
In the section Negative-number solutions it says
These have only two terms on the right side, so what are x, y, and z in each case? Loraof ( talk) 16:10, 18 February 2016 (UTC)
It would be nice to give a table or list of numerical examples—each solution for, say, n up to 10 or 15. Loraof ( talk) 01:14, 8 July 2017 (UTC)
AirdishStraus ( talk) 15:23, 8 July 2017 (UTC)
فرمول این حدس را پیدا کرده ام و این حدس کاملا درست است Mohammad ghanbary ( talk) 21:45, 16 January 2021 (UTC)
I solve it Mohammad ghanbary ( talk) 22:09, 16 January 2021 (UTC)
GA toolbox |
---|
Reviewing |
Reviewer: HenryCrun15 ( talk · contribs) 04:27, 8 January 2022 (UTC)
Hi, I will review this article. HenryCrun15 ( talk) 04:27, 8 January 2022 (UTC)
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 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:
Background:
Modular identities:
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 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. |
| |
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. |
| |
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:
@ 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)
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: | ||
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. | |
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: | ||
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. | |
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 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: | ||
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:
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. | |
3b. it stays focused on the topic without going into unnecessary detail (see summary style). | All good. | |
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. | 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)