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== Heronian triangle triples with distinct side lengths ==

Why does the section on Heronian triangle triples in [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&oldid=1014416856#Heronian_triangle_triples the present article] focus on Heronian triangles with distinct side lengths; that is with the property that no two lengths are equal?  This is not a focus of the main article on [[Heronian triangle]]s.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 23:29, 26 March 2021 (UTC)
:I have edited that section of the article so that the emphasis is not on distinct side lengths. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 17:35, 15 January 2022 (UTC)

== RfC: Should an extended definition reside only in the lead? ==
{{closed rfc top
| status = 
| result = There is a general consensus to keep the lead along the lines of version A as opposed to version B, in particular to include a definition of primitive Pythagorean triple in the lead. {{nac}} ([[User talk:Buidhe|t]] &#183; [[Special:Contributions/Buidhe|c]]) '''[[User:buidhe|<span style="color: black">buidhe</span>]]''' 22:18, 26 April 2021 (UTC)
}}

Should Pythagorean triples, primitive Pythagorean triples, and multiples of primitives, be defined only in the lead paragraph, as in [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&oldid=1016805042 this version] '''A'''. Or should the extended definition be in the body of the article, and a less technical version reside in the lead paragraph, as in [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&oldid=1016796212 this version] '''B'''.

Relevant guidelines are [[MOS:LEAD]] and [[WP:TECHNICAL]].

[[User:Lawrencekhoo|LK]] ([[User talk:Lawrencekhoo|talk]]) 02:46, 12 April 2021 (UTC)

===Comments===
*I don't think primitive pythagorean triples should be in the first paragraph, but I believe we should still define them in the lede. Much of the article is about primitive pythagorean triples and isn't applicable to the general case. It seems that primitive pythagorean triples are of much greater interest than regular ones. [[User:Chess|Chess]] ([[User talk:Chess|talk]]) <small>(please use&#32;{{tlx|reply to|Chess}} on reply)</small><!--Template:Please ping--> 04:42, 12 April 2021 (UTC)

*The statement of the RFC is inaccurate. It states that "a less technical version" of the definition of primitive Pythagorean triples exists in the lead in version B. In fact, version B completely omits primitive Pythagorean triples from the lead altogether, and makes other important changes outside the lead, replacing the important "examples" section by a very-stubby "definitions" section with examples shoehorned into it more to fill it out than because they make sense to be in the same section. More, this RFC appears to be "I didn't get my way and I am going to raise a fuss until I do" rather than an actual attempt to ''first'' discuss what reasonable alternatives might exist and use that discussion to collaboratively develop improved wording and ''only once that has been done'' assess the levels of support for those alternatives. The statement of the RFC doesn't make a clear distinction between "we should follow these general principles" and "we should use this specific version of the article inaccurately described as being the one that follows these general principles" and I think it is impossible for such an RFC to create a non-muddled mandate for change. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 07:41, 12 April 2021 (UTC)

*This RFC is essentially a continuation of the discussion of the preceding section, where I opposed the exact formulation of [[MOS:LEAD]] to its interpretation by the opener of this RfC. So, I agree with the comment by {{u|David Eppstein}}, and in particular with {{tqq|this RFC appears to be "I didn't get my way and I am going to raise a fuss until I do"}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:17, 12 April 2021 (UTC)
:* I do not appreciate your disparaging my motives. Kindly explain what an editor should do instead of starting an RfC if he gets into a dispute about content with an editor unwilling to budge? Be honest, D.Lazard – how likely was it that you would not have  reverted any further attempts of mine to edit the lead? Especially after brushing off my objections as nothing.  [[User:Lawrencekhoo|LK]] ([[User talk:Lawrencekhoo|talk]]) 02:59, 13 April 2021 (UTC)

*To be blunt, this looks like a "I didn't get my way so I'm going to involve as many people as I can until I do" situation. Don't do that, {{u|Lawrencekhoo}}. Along with that, I agree with {{u|David Eppstein}} and {{u|D.Lazard}}. —[[User:Moonythedwarf|moonythedwarf]] (Braden N.) 12:57, 12 April 2021 (UTC):

*Version A reads better and is in line with the relevant guidelines. I fail to see how version B would be an improvement. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 18:27, 12 April 2021 (UTC)

* To be clear, I'm not married to version B. The main thing I would like to see is an easier to read first paragraph, and a more extended definition in the body of the article. [[User:Lawrencekhoo|LK]] ([[User talk:Lawrencekhoo|talk]]) 02:59, 13 April 2021 (UTC)
** I'm still not seeing what the difficulty with the first paragraph is supposed to be. It hardly seems less approachable than the intro of [[Hilbert space]], which you mentioned in the previous section as an example to follow. That drops "vector algebra" into the second sentence, and by the third it's talking about {{tq|a vector space equipped with an inner product}}. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 03:32, 14 April 2021 (UTC)
* Definitely not A. It's hard to read and technical. Move it to the body of the article. [[User:Darx9url|Darx9url]] ([[User talk:Darx9url|talk]]) 03:08, 15 April 2021 (UTC)

* The definitions of '''Pythagorean triple''' (PT) and '''primitive Pythagorean triple''' (PPT) are so simple, I don't see any point in having a separate Definitions section for them. The concept of a PPT is so fundamental to the study of PTs, it occurs everywhere in this article; hence it should be defined in the lead. Since both definitions are so short, I don't see any good reason not to define both in the first paragraph. Similarly for '''Pythagorean triangle'''.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 04:21, 15 April 2021 (UTC)
** This pretty much sums up my feelings, too. The basic concept is technical; you can't take the math out of the lead when the article is about math. And unlike the example of [[Hilbert space]], the definitions are short, so they don't need a whole section of motivation and explication. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 14:56, 15 April 2021 (UTC)
{{closed rfc bottom}}

== Si.427 ==

His article needs more history, not less. Contrary to {{u|David Eppstein}}'s claim, the link https://doi.org/10.1007%2Fs10699-021-09806-0 does work. Si.427 shows a survey of plots of land laid out with exceptional precision and explains that the triples (5, 12, 13) and (8, 15, 17) were used because the plots are more narrow than the customary (3,4,5) triple allows. As a very early use of Pythagorean triples, this is ''obviously'' relevant to the article. Actually this article should have a whole section on the ancient history, in which the Babylonian knowledge and application would have a starring role. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 09:45, 6 August 2021 (UTC)
:I reverted it back in while we have a discussion, my understanding was that triples were definitely a part of this "discovery", I might be wrong, let's see.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 11:50, 6 August 2021 (UTC)
::The tablet makes no such explanation. The idea that those triangles were used because of being Pythagorean is a modern interpolation. The existence of those triangles in the layout could plausibly be total coincidence. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:11, 6 August 2021 (UTC)
:Whether the doi 'link' works probably depends on your browser. It's a unique identifier for the article, not a a web URL, so if it does work it's because your browser does some work behind the scenes. You can find the article at https://link.springer.com/article/10.1007/s10699-021-09806-0, which is a web URL. It definitely mentions Pythagorean triples. We can debate whether it's notable enough to be included in the lede. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 17:44, 6 August 2021 (UTC)
::: This is not correct. https://doi.org/10.1007%2Fs10699-021-09806-0 is an ordinary web url that the browser doesn't need to know anything about. The work of redirecting it to the object with that DOI is done by doi.org. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 13:18, 7 August 2021 (UTC)
::The lead already speaks of Plimpton 322. There is this earlier paper as well https://www.researchgate.net/publication/342084463_Perpendicular_Lines_and_Diagonal_Triples_in_Old_Babylonian_Surveying Still, the lead ought to reflect something in the body (usually). It seems deserving enough for the body after a brief inspection, the lead I don't know.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 18:01, 6 August 2021 (UTC)
::A couple of RS backing up the story https://www.sciencefocus.com/news/the-babylonians-were-using-pythagoras-theorem-over-1000-years-before-he-was-born/ and https://www.theguardian.com/science/2021/aug/05/australian-mathematician-discovers-applied-geometry-engraved-on-3700-year-old-tablet admittedly this is mainly story not math but the idea of using a right triangle to get a right angle/normal seems clear enough.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 10:48, 7 August 2021 (UTC)
::: Not being able to read cuneiform I can't say what the tablet literally says. However, we are supposed to take the advice of the experts who publish on the subject. Here we have a peer-reviewed article which concludes that pythagorean triplets were explicitly known about and it cites earlier scholarly research that reached the same conclusion. So it is relevant for the article. If we find RS that doubt the conclusion we should of course cite them too. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 13:24, 7 August 2021 (UTC)
::::We should report what the consensus of scholars, as reflected in secondary sources, is. This is a primary source and too soon for us to have a clear sense of how it has been accepted by others. As such, it is not required that we rush to breathlessly include all its speculation in our article, stated as unquestioned and established fact. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:34, 7 August 2021 (UTC)
:::::I think you are going a bit beyond your remit, WP reports what reliable sources say, WP editors don't get to decide what is speculative, do you have a source that says it is speculative? I have seen quite a few rs discussing this and I didn't see the word speculative anywhere. We can also attribute if you are squeamish about stating it as fact in WP voice, I wouldn't have a problem with that myself.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 17:17, 7 August 2021 (UTC)
::::::I think attribbution to Daniel Mansfield from the [[University of New South Wales]] is a good idea. Also {{re|David Eppstein}} kind of puzzled by [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&type=revision&diff=1037446034&oldid=1037412962 FAILED VERIFICATION], the link works for me, there are also more sources in [[Si.427]]. [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 17:36, 7 August 2021 (UTC)
:::::::When it was first added I got an error page from Springer when I tried following it, and the paper didn't show up at all in Google Scholar searches. Seems to be ok now. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:41, 7 August 2021 (UTC)
::::::::{{re|David Eppstein}} would you object inclusion with attribbution to Daniel Mansfield? [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 17:45, 7 August 2021 (UTC)
:::::::::It depends on what you mean by "inclusion". If you mean something asserting that Si427 definitely has Pythagorean triples on it, and that they were definitely used for surveying, then yes, I object. If you mean something like "Si 427, a tablet depicting the survey of a plot of land, differs from many other such tablets in showing rectangular plots of land rather than non-rectangular quadrilaterals. Daniel Mansfield has speculated that the sizes of these plots are similar to those derived from Pythagorean triangles other than the 3-4-5 triangle, and that this suggests that these triangles were known to the Babylonians and used for surveying", then I think it might be an accurate summary of what the paper says. It is still a rush to report primary-sourced research, without any secondary sourcing to give us any indication to tell us how this speculation has been received by experts in this area, so I would still prefer to leave it out for now, but at least it wouldn't be as objectionable. I'd add that I find Mansfield/Wildberger's approach to doing science by press release, the accompanying push to include that material into Wikipedia every time it happens, and the appearance of new editors to make that push, promotional and concerning. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 17:51, 7 August 2021 (UTC)
::::::::::Agreed. A little bit more explanation when you revert a good faith addition would go a long way however. This page had been on my watch list for a while, and I wasn't aware of the Mansfield - Wildberger collaboration, nor of their aggressive habit of issuing promotional press releases. So when I saw the summary reversion of material backed by an apparently legitimate scholarly publication that drew my attention. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 18:04, 7 August 2021 (UTC)
:::::::::::Apparently you can't win. If you provide in-depth edit summaries like "How about using real scholarship on this topic rather than WP:REFSPAMming WIldberger cruft?" and "That link doesn't work, but https://doi.org/10.1086/709309 by the same author is clear that Si427 is either an actual land survey or an exercise in land surveying, not about Pythagorean triangles except coincidentally" you get told that your edits would have been fine if only you had provided an edit summary that was more detailed, but if you provide in-depth edit summaries like "Your new shortdesc violates WP:SDNOTDEF by providing zero information beyond what is already in the title. Don't do that." you get the target of the summary [https://en.wikipedia.org/w/index.php?title=User_talk%3ADavid_Eppstein&type=revision&diff=1037234963&oldid=1037189341 whining on your talk page] that "what you wrote was tactless and unconstructive, and I think that was a poor way of handling what was a simple mistake". My impression is that people just don't like their bad edits being undone and will whine about it regardless of how clearly you explain why they were bad. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 21:30, 7 August 2021 (UTC)
::::::::::::Since the author of the article was Mansfield, not Wildberger, it wasn't clear how this amounted to refspamming Wildberger cruft, unless you were already aware of the Wildberger - Mansfield collaboration. Also, the insertion was not my work, and I just agreed with you (now that I have a fuller picture) that the article shouldn't be mentioned. So yes, you can actually 'win' if you explain things a bit better. Bold edits are fine, so are reverts, and a little bit of tact will go a long way towards a productive discussion. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 00:02, 8 August 2021 (UTC)
::::::::::Idk who you mean by new editors but I trust you are not referring to me. I used to get involved in math articles on here a while back and there is a reason I don't bother much with it now. The history of mathematics rather than mathematics per se, is what I am interested in here. I am not here to push someone's pet theory, yours included.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 18:07, 7 August 2021 (UTC)
{{od}}
I concur with {{u|David Eppstein}}'s statement that {{tq|We should report what the consensus of scholars, as reflected in secondary sources, is. This is a primary source and too soon for us to have a clear sense of how it has been accepted by others. As such, it is not required that we rush to breathlessly include all its speculation in our article, stated as unquestioned and established fact.}} The ''Guardian'' article is downright irresponsible for not getting [https://twitter.com/Eleanor_Robson/status/1422960389231349761 commentary from archaeological experts], who'll see their "oldest known example of applied geometry" and raise them the pyramids of Giza. It also uncritically parrots Mansfield and Wildberger's claims about Plimpton 322, despite [https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/ even mathematicians not buying it]: {{tq|For one, the tablet contains some well-known errors, so claims that it is the most accurate or exact trig table ever are just not true.}} The ''Science Focus'' item is equally superficial and comparably bad. [[Silly season]] hype in general-interest newspapers is not a suitable basis for an encyclopedia article. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 18:57, 7 August 2021 (UTC)
:Wonderful, WP:SCHOLARSHIP backed up by WP:NEWSORG refuted by blog.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 22:17, 7 August 2021 (UTC)
::Your failure to recognize Robson's name and expertise in this subject, and your failure to apply the "recognized expert" clause of [[WP:SPS]] does not help your cause. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 23:19, 7 August 2021 (UTC)
::A little bit more commentary about this article by Viktor Blåsjö, a lecturer at Utrecht University who specialises in the history of mathematics: https://twitter.com/viktorblasjo/status/1423417354269171716. He says the media hype over this is absurd. He also engages with Robson's criticism, partially agreeing with her, and partly disagreeing, but not in a way that argues in favour of the notability of this article. It is potentially fine for inclusion in some Wikipedia article at some stage, after other scholars have had the time to react to it, provided it then meets the notability test, which right now it doesn't. Nothing wrong with boldly inserting this article here, and challenging the grounds for reverting it, but right now it just doesn't belong in the article on Pythagorean triples. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 00:14, 8 August 2021 (UTC)
:::Note also new Wikipedia article [[Si.427]] where Infinity Knight has been uncritically repeating the same hyped-up claims. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:39, 8 August 2021 (UTC)
:::: I think that it is bizarre, nay false, to claim that a journal article by Daniel Mansfield is a primary source. Nobody is proposing to cite Si.427 itself. I agree entirely that we should stay away from the hype that appears in newspapers, but the debate between scholars is pertinent and citable. If there's a debate, we report it as a debate. This is very elementary wikicraft. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 01:49, 8 August 2021 (UTC)
::::: A journal article by Mansfield is self-evidently a primary source for any claims by Mansfield. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 05:20, 8 August 2021 (UTC)
::::::Yes. Si.427 itself is a land survey. Unlike some other tablets the mathematical content is not self-verifying, but rests on quite a lot of interpretation that cannot be found on the tablet itself. The Mansfield paper is primary for that interpretation, because it can be found nowhere else. Even for some other tablets like [[YBC 7289]] where the mathematical content is obvious and easily read, there is still room for interpretation (that it was student work). Here, the interpretation is the only thing that makes it even plausibly relevant for this article. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 06:43, 8 August 2021 (UTC)
::::::I note that WP policies are being rewritten by random wp editors with an opinion. A peer reviewed paper has evidence-free assertions made about the content. Well, I will just keep adding sources for all those rs apparently being duped:
https://www.newscientist.com/article/2285917-babylonians-calculated-with-triangles-centuries-before-pythagoras/ (not the blog section of NS, actual NS)
https://www.smithsonianmag.com/smart-news/ancient-tablet-shows-babylonians-used-pythagorean-geometry-1000-years-pythagoras-180978376/
https://www.economist.com/science-and-technology/the-babylonians-used-pythagorean-ideas-long-before-pythagoras/21803301
https://theconversation.com/how-ancient-babylonian-land-surveyors-developed-a-unique-form-of-trigonometry-1-000-years-before-the-greeks-163428
and we mustn't forget the university itself, clearly staffed by fools:
https://news.unsw.edu.au/en/australian-mathematician-reveals-oldest-applied-geometry
The only smart chap is the fellow on the blog. [[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 10:12, 8 August 2021 (UTC)
: "self-evidently a primary source for any claims by Mansfield", "Mansfield paper is primary for that interpretation". By that logic, every academic paper with an original thought is primary for that thought, and (therefore? by what policy?) can't be cited?? Half of Wikipedia vanishes in a puff of smoke!! Even if we adopt the false claim (by WP's meaning of the term) that an original idea maketh a primary source, there is still no policy against citing it. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 12:11, 8 August 2021 (UTC)
:: No, this is just the standard we have for any academic topic: the source where an idea originates is a primary source for that idea. We've been covering math and science in this way for years now, relying on the later analyses in places like textbooks and review articles to establish the necessary perspective. Wikipedia's not going up in a wildfire. On the contrary, if we didn't hold ourselves to this standard, we'd be buried under new "interpretations" of historical artifacts that were published once and then ignored by the scholarly community ever after. [[WP:NOT|This is an encyclopedia, not an indiscriminate pile of factoids]].
:: A news release from the University of New South Wales is not an [[WP:INDY|independent]] source. It's a salvo in a publicity campaign. Universities do that all the time. Likewise, the item in ''The Conversation'' is an essay by Mansfield himself. None of the "news" stories amount to [[WP:SIGCOV|significant coverage]], barely rising above [[churnalism]]. They take materials provided by Mansfield and UNSW, lightly reprocess them, uncritically repeat old claims about Plimpton 322, and fail even to include the typical quote from "an expert not involved in the research". Not a one of them is helpful for our purposes. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 17:26, 8 August 2021 (UTC)
::: So this is Mansfield's interpretation reported by a large number of secondary sources, see above. If there are opposing scholar interpretations those should be included as well. Still, {{green|For one, the tablet contains some well-known errors, so claims that it is the most accurate or exact trig table ever are just not true.}}    [https://blogs.scientificamerican.com/roots-of-unity/dont-fall-for-babylonian-trigonometry-hype/ here] does not impress me, since (a) Mansfield clearly addresses those commonly agreed upon errors, see '''Table 8 Translation of Plimpton 322, with errors underlined''' [https://link.springer.com/article/10.1007%2Fs10699-021-09806-0 here] and (b) uses "accurate" in terms of Babylonian reciprocal table calculation technique, i.e. without approximation of sin()/cos() trig. [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 10:38, 10 August 2021 (UTC)
::::You may well be right but do not fall into the same trap as others here, we need reliable sources to make such comments.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 11:12, 10 August 2021 (UTC)
:::::My point is there are no reliable opposing scholar interpretations for Si.427 right now, despite wide rs coverage of Mansfield's interpretation. [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 11:27, 10 August 2021 (UTC)
::::::You are deliberately repeating a falsehood here. Both Robson and Blåsjö meet the "recognized expert" clause of [[WP:SPS]]. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 16:25, 10 August 2021 (UTC)
:::::::{{green|deliberately repeating a falsehood}}?!? it is interesting what kind of Si.427 interpretations could one extract from those tweets 🤦‍♂️ It least [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&type=revision&diff=1037446034&oldid=1037412962 FAILED VERIFICATION] is no longer a reason for your objection. [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 04:53, 12 August 2021 (UTC)
::::::Opposing interpretations are not necessary; I don't think any responsible scholar is going to come up with a rival interpretation on the fly in response to some press coverage. What we have here is a methodological critique, and  Robson's tweets make clear that she believes the methodological critique made in her late nineties/early aughts articles and books of much of past scholarship on Mesopotamian mathematics apply to this work as well. And her critique is very much in the spirit of Unguru's earlier and highly influential critique of past practices in history of mathematics more generally. Interestingly Blåsjö is known to be critical of the Unguru/Robson "consensus" but agrees that press coverage here has been highly misleading. In my own experience, based on what little work I have done on Wikipedia's history of mathematics articles, there is no substitute for a careful reading of the work of professional historians. What one finds in the press, popular mathematics books, mathematics textbooks, and even history of mathematics textbooks is likely to be superficial and inaccurate. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 11:36, 12 August 2021 (UTC)
:::::::The article [[Plimpton 322]] exists and criticisms of it belong there and so does [[Si.427]] and if and when any criticism of it appears then it can go there. This is archaeology, folks, not math, there's no "proof" of anything.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 12:03, 12 August 2021 (UTC)

===RfC===
{{closed rfc top
| status = 
| result = Closing early as badly formed question and per [[WP:SNOW]].

RfCs are not really useful for determining statements of fact, nor is that their purpose. As to the question at hand, there's a clear consensus (which wasn't really necessary) that the source in question ''is'' indeed a primary source, and that this is not the only concern which should guide further discussion of this, such as the policy on neutrality and giving the appropriate amount of prominence to opposing viewpoints (as well as favouring the academic mainstream, whatever that may be in this situation). [[User:RandomCanadian|RandomCanadian]] ([[User talk:RandomCanadian|talk]] / [[Special:Contributions/RandomCanadian|contribs]])  02:21, 10 August 2021 (UTC)
}}


Per the discussion above (RFCBEFORE), in relation to this article, is the peer reviewed paper {{cite journal | author = Daniel F. Mansfield | title = Plimpton 322: A Study of Rectangles | journal = Foundations of Science |url=https://link.springer.com/article/10.1007/s10699-021-09806-0| year = 2021}} a primary source? [[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 21:48, 8 August 2021 (UTC)
====Discussion====
*'''No''' Per [[WP:SCHOLARSHIP]]. The paper is also cited by reliable sources, [[WP:NEWSORG]]. Subject matter includes material re [[Plimpton 322]] and [[Si.427]] therefore notable.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 21:52, 8 August 2021 (UTC)
*'''Yes, it is a primary source''', for the claims relevant for this article. (The RFC is technically badly formulated, as the primary or secondary nature of a source depends on what you're using it for, but the source would be irrelevant if only used for its secondary claims on the land-survey nature of the tablet.) The popular-press coverage based on press releases is secondary, but not independent and not reliable (because it contains no effort to obtain independent opinions from experts in this area). And as I said above, the persistence of the Mansfield/Wildberger press-release-pushers here, despite no declared conflicts of interest but in the face of dismissals of the significance of this work from experts including Robson and Blåsjö, is very concerning to me. That attitude does not reflect the [[WP:NPOV]] requirement of sticking to the consensus of mainstream views on contentious topics and, while it may be motivated by good faith desires to reflect new scholarship rather than pure promotionalism, it does not do any service to our readers in guiding them to that consensus view. Escalating the discussion to an RFC is a symptom of that misguided fervor to push this material into Wikipedia. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 22:09, 8 August 2021 (UTC)
*'''Yes,''' it is a primary source, as far as the claims for which it is here being invoked are concerned. Being the beneficiary of an obvious (and probably well-funded) publicity campaign doesn't change that fact. [[WP:SCHOLARSHIP]], mentioned above, says the following: {{tq|For example, a paper reviewing existing research, a review article, monograph, or textbook is often better than a primary research paper.}} The paper we're discussing here is a {{tq|primary research paper}}, not a review article, a monograph, or a textbook. Perhaps some confusion may be stemming from the fact that it is not a historical inscription itself, but that's not what Wikipedia policies and guidelines mean by "primary source". For analogy, a journal article on the ''Odyssey'' that advances a new hypothesis about what inspired the episode of the [[lotus-eaters]] would be a primary source for that idea, even though the poem is [[Odyssey#Textual_history|not exactly new]]. The [[WP:SCHOLARSHIP]] guideline also advises us to beware of low-quality journals. ''Foundations of Science'' appears to deal more in philosophy of science than anything else; its [https://www.springer.com/journal/10699/editors editorial board] has experience in psychology, philosophy, and in quantum mechanics, but not archaeology or the history of mathematics. Honestly, it seems a strange place to publish a paper like this one. The journal may be fine for many purposes, but I have to wonder how good they are at finding qualified reviewers in this area. This underlines the importance of waiting for ''scholarly'' feedback before diving in and writing material based on what the university's PR department puts out. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 23:34, 8 August 2021 (UTC)
*'''Yes''' - This particular source is a primary paper.  "The paper is also cited by reliable sources" does not change the status of this source, but may be useful to evaluate if the material is [[WP:DUE]] (and those secondary sources can be useful for independent coverage when they exist).  —[[User:PaleoNeonate|<span style="font-variant:small-caps;color:#44a;text-shadow:2px 2px 3px DimGray;">Paleo</span>]][[User talk:PaleoNeonate|<span style="font-variant:small-caps;color:#272;text-shadow:2px 2px 3px DimGray;">Neonate</span>]] – 01:25, 9 August 2021 (UTC)
*'''Yes,''' it is a primary source. For many of the same reasons noted above, but in addition because it was peer reviewed and has reliable citations. [[User:Jurisdicta|Jurisdicta]] ([[User talk:Jurisdicta|talk]]) 06:09, 9 August 2021 (UTC)
*'''Yes''', for the reasons stated above by David Eppstein. That by itself doesn't mean we cannot cite it. Blåsjö too says that this time round the article itself looks fine, and even defends it against criticism by Robson, though he agrees with the criticism of the hype around it. But given that this is a very recent article arguing for a new hypothesis, we should wait for reactions in scholarly journals before citing it here. Citing it right now would mean giving it [[WP:UNDUE]] attention. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 10:59, 9 August 2021 (UTC)
*'''It's the wrong question''' since there is no rule against citing primary sources anyway. Peer reviewed academic studies are the '''gold standard''' for sourcing on Wikipedia and only the tiny fraction that are pure surveys cannot be called primary sources if the definition is pressed literally. The only ''useful'' assignment of roles in this instance is that Si.427 is the primary source and the paper about it is a secondary source. The claim that we are entitled to dissect such sources and treat them differently if they have original ideas is a claim totally unsupported by policy. Actually it is OR and thus against policy. It is a fact that all across the encyclopedia original peer-reviewed research is cited in articles. Even some first-rate original research by David Eppstein. [[User:Zero0000|Zero]]<sup><small>[[User_talk:Zero0000|talk]]</small></sup> 12:14, 9 August 2021 (UTC)
*: {{tq|Peer reviewed academic studies are the '''gold standard''' for sourcing on Wikipedia}} Sorry but this is plainly false; see for example the second paragraph at [[WP:MEDRS]]. --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 16:27, 9 August 2021 (UTC)
*: I was about to make the same comment. Getting through peer review isn't by itself the "gold standard". [[WP:SCHOLARSHIP]], mentioned above, makes repeated statements to this effect. Recognizing which parts of a reference are new ideas and which are old isn't OR; it's just part of reading that reference carefully, a necessary step in doing anything with it. Nor does rightfully designating a source as primary forbid it from being used. "Primary" doesn't mean "bad". [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 16:47, 9 August 2021 (UTC)
*::It's up to reliable sources to read the material not WP editors, the latter being OR (a lot of that on this page).[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 16:57, 9 August 2021 (UTC)
*:::OR is saying things in articles that are not said in sources. This is not that. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 19:04, 9 August 2021 (UTC)
*'''Wrong question''', as [[WP:PRIMARY]] allows primary sources, with caution, and very frequently does use them. [[User:MarshallKe|MarshallKe]] ([[User talk:MarshallKe|talk]]) 22:12, 9 August 2021 (UTC)
{{closed rfc bottom}}

== Babylonian mathematics ==

For information, I added the following to the [[Babylonian mathematics]] article which has a section on Plimpton 322:

According to Dr.Daniel Mansfield, a researcher at [[University of New South Wales|UNSW Sydney]],<ref>{{cite web|title=Dr. Daniel Francis Mansfield|url=https://research.unsw.edu.au/people/dr-daniel-francis-mansfield|access-date= 6 August 2021}}</ref> speaking about the "surveyor's tablet" [[Si.427]] and its relationship with [[Plimpton 322]], "The rectangles are perfect". The surveyor achieved this by using Pythagorean triples.<ref>{{cite news|title= Babylonians calculated with triangles centuries before Pythagoras|url= https://www.newscientist.com/article/2285917-babylonians-calculated-with-triangles-centuries-before-pythagoras/|date=4 August 2021|access-date=7 August 2021}}</ref><ref>{{cite journal | author = Daniel F. Mansfield | title = Plimpton 322: A Study of Rectangles | journal = Foundations of Science | year = 2021 | doi = 10.1007/s10699-021-09806-0 }}</ref><ref>{{Cite web|url=https://news.unsw.edu.au/en/australian-mathematician-reveals-oldest-applied-geometry|title=Australian Mathematician reveals oldest applied geometry|website=news.unsw.edu.au}}</ref>

[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 10:28, 9 August 2021 (UTC)
:Sorry, but 2 out of the 3 sources for the actual claim there are press release/[[churnalism]]. This looks [[WP:UNDUE|undue]] for a broad survey article, which should only state the most essential and solidly established material about Plimpton 322. It's also not particularly informative: what are we supposed to get out of "The rectangles are perfect"? It's a dramatic statement, but not an educational one. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 16:50, 9 August 2021 (UTC)
::No need to apologize.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 16:53, 9 August 2021 (UTC)
<!-- [[User:DoNotArchiveUntil]] 10:01, 14 September 2021 (UTC) -->{{User:ClueBot III/DoNotArchiveUntil|1631613668}}
An RFC (above) has determined that the paper at Reference 3 below should be considered as a primary source. Should the following (or something similar or anything at all) be included in the article body?
According to Dr.Daniel Mansfield, a researcher at [[University of New South Wales|UNSW Sydney]],<ref>{{cite web|title=Dr. Daniel Francis Mansfield|url=https://research.unsw.edu.au/people/dr-daniel-francis-mansfield|access-date= 6 August 2021}}</ref> speaking about the "surveyor's tablet" [[Si.427]] and its relationship with [[Plimpton 322]], "The rectangles are perfect". The surveyor achieved this by using Pythagorean triples.<ref>{{cite news|title= Babylonians calculated with triangles centuries before Pythagoras|url= https://www.newscientist.com/article/2285917-babylonians-calculated-with-triangles-centuries-before-pythagoras/|date=4 August 2021|access-date=7 August 2021}}</ref><ref>{{cite journal | author = Daniel F. Mansfield | title = Plimpton 322: A Study of Rectangles | journal = Foundations of Science | year = 2021 | doi = 10.1007/s10699-021-09806-0 }}</ref><ref>{{Cite web|url=https://news.unsw.edu.au/en/australian-mathematician-reveals-oldest-applied-geometry|title=Australian Mathematician reveals oldest applied geometry|website=news.unsw.edu.au}}</ref>
[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 09:10, 10 August 2021 (UTC)

{{reftalk}}

====Discussion====
*'''Yes''' Per [[WP:SCHOLARSHIP]]. The paper is also cited by reliable sources, [[WP:NEWSORG]]. Subject matter includes material re [[Plimpton 322]] and [[Si.427]] therefore notable. I have not been able to locate criticism of the paper in any reliable source. Should any emerge, it can be added. Two short sentences in the article, which already references Plimpton 322 in its lead, does not seem undue.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 09:16, 10 August 2021 (UTC)
*'''No''', per [[WP:UNDUE]]. The source itself is fine, but this is a very recent paper, advancing a new hypothesis. We should wait for other peer-reviewed papers to engage with it. Secondarily, we should not be taken in by a publicity campaign to get this included, especially when some of the Wikipedians involved are collaborating with the author of the article to get it mentioned. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 10:37, 10 August 2021 (UTC)
*:In case that last evidence-free allegation was directed at myself, for the record, I do not know and have never heard of Mansfield until the discussion arose in respect of the recent edit.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 11:26, 10 August 2021 (UTC)
*'''No.''' Almost every new interpretation of math. in cuneiform tablets leads to controversies. So, per [[WP:UNDUE]], it is unencyclopedic to mention an interpretation that has not yet been discussed by other experts. This is exactly the reason for which care is needed for using primary sources. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 17:11, 10 August 2021 (UTC)
*'''No,''' per [[WP:UNDUE]]. As has been pointed out above, the "news" stories being provided did not actually engage in analysis or investigation; to all appearances, they were written based on the university's own PR material. In the absence of ''scholarly'' feedback, our most responsible course of action is to say nothing. The suggested sentence also violates Wikipedia house style regarding [[MOS:PHD|honorifics]] and seems to be written more for drama than for imparting information. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 18:03, 10 August 2021 (UTC)
*'''No'''. Among other reasons already discussed above, the suggested wording phrases the sentence "The surveyor achieved this by using Pythagorean triples." as a statement of fact, stated in the voice of Wikipedia, when in fact it is the opinion of Mansfield from the paper and we have no clear information from a consensus of scholarly sources whether it is a new opinion, or an established one, or a fringe theory, and whether scholars believe this supposed use of non-3-4-5 Pythagorean triangles to be widespread or whether this is an unusual instance. The added wording does not even clearly distinguish this case (where non-3-4-5 triangles are claimed to have been used) from the well-established use of 3-4-5 triangles for this purpose. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 18:16, 10 August 2021 (UTC)
::The article begins with an extensive summary of the relevant scholarship to date, and is properly sourced (obviously, or it would not have passed peer review). It might therefore make a decent secondary source about the current state of scholarship (excluding the new hypothesis). But even then we should probably wait to see if other scholars use it at such by citing it approvingly. [[User:Mmeijeri|Martijn Meijering]] ([[User talk:Mmeijeri|talk]]) 19:02, 10 August 2021 (UTC)
::The sentence "The surveyor achieved this by using Pythagorean triples." is stated by NS in their voice and separately from a quote by Mansfield. I am not given to source misrepresentation.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 22:11, 10 August 2021 (UTC)
::Another respectable RS, [[Deutsche Welle]] (DW) [https://www.dw.com/en/thank-the-babylonians-not-pythagoras-for-trigonometry/a-58820147 Thank the Babylonians not Pythagoras for trigonometry] [[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 22:17, 10 August 2021 (UTC)
:::Why are you still citing press-release churnalism rather than proper scholarship, and calling it "respectable"? —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 22:24, 10 August 2021 (UTC)
:::{{ec}} Like every other "news" source provided so far, that one quotes no one but Mansfield. It's PR, all the way down. It also makes statements that are downright nonsensical, like {{tq|Instead of using angles and circular or trigonometric functions, such as the Greeks did, the Babylonians based their calculations on numerical relationships.}} How are trigonometric functions ''not'' "numerical relationships"? What is that even trying to say? Who knows? It also implicitly gives Mansfield credit for the discovery that the Babylonians knew about Pythagorean triples, and the interpretation of them as trigonometric; needless to say, that interpretation is [[Plimpton_322#Purpose_and_authorship|decades old]]. It also says, {{tq|Until now we've gone with the story that trigonometry was invented by the ancient Greeks and more specifically by the philosopher and mathematician Pythagoras.}} The Greek credited as the "father of trigonometry" is [[Hipparchus]]; no serious historian of mathematics would make a definitive statement about what Pythagoras did or didn't do, given that [[Pythagoras#Biographical_sources|exactly zero writings by the man himself survive, and the later sources contradict one another]]. Let's let the wave of publicity pass and wait for serious commentaries to be written. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 22:38, 10 August 2021 (UTC)
::::I have cited the scholarship that there is (a primary source about a primary source), you just don't like it? If there were any thing else worth citing I would cite that too, it's in the Daily Mail, I left that one out. If there's an RS with something criticizing the article I am sure we will hear all about it in short order.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 22:45, 10 August 2021 (UTC)
:::::The problem isn't that I "just don't like it"; the problem is that it's ''not scholarship.'' It's clickbait. This is what you get when news media, who can be serious on other topics, get a slick press package about a subject where they have no in-house experience and don't care enough to call around for commentary. [[User:XOR&#39;easter|XOR&#39;easter]] ([[User talk:XOR&#39;easter|talk]]) 22:55, 10 August 2021 (UTC)
::::::You learn something every day.[[User:Selfstudier|Selfstudier]] ([[User talk:Selfstudier|talk]]) 22:59, 10 August 2021 (UTC)
::::Attributing the invention of trigonometry to Pythagoras??? You would think history of mathematics coverage in the press couldn't get any worse; then it hits a new low. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 12:08, 12 August 2021 (UTC)
*'''Yes''' Per [[User:Selfstudier|Selfstudier]] [[User:Infinity Knight|Infinity Knight]] ([[User talk:Infinity Knight|talk]]) 07:28, 12 August 2021 (UTC)
*'''No'''. I agree with the comments of [[User:D.Lazard|D.Lazard]] and [[User:XOR&#39;easter|XOR&#39;easter]]. The proposed text was earlier added to the short (and appropriately so) section on Plimpton 322 in [[Babylonian mathematics]], resulting in unbalanced coverage; it was quickly deleted for that reason. The coverage of Plimpton 322 in the present article is even more brief; hence the imbalance problem that would result from adding the proposed text here is even more acute. [[User:Will Orrick|Will Orrick]] ([[User talk:Will Orrick|talk]]) 12:01, 12 August 2021 (UTC)

== Lem Chastain comments ==
I just glanced at all of the "Pythagorean Triplets" sites that came up.  I was surprised that Euclid's Formula seems to be the latest word on this topic.  The most innovative switched the parameters 'm' and 'n' to some other letters!
In sum, the sites all repeated the same stuff.  [ljc -- So, nothing new in three millennia.]

I don't have time to write an update of this site now -- based on my look at this problem years ago -- as it would take hours and my notes are not in draft form.  [Don't take this as a 'note on the margin of the page' like a comment about a solution to one famous problem; which solution was never found in that mathematician's papers.]

Further, and more importantly, I got nowhere in trying to add to the "Axonometric Projections" topic-- except for my comments at the end of the 'Talk' section; so won't make an effort here unless I here that someone is interested.  (I couldn't change that article without citing a source, but my own article in 1989 was the source.  You will now find that a DOI for my correction in APA style is included in that comment section; which DOI was issued by the original journal: Engineering Design Graphics Journal.

It isn't hard to generate a table of solutions, based on two parameters.  I initially mentioned my idea to H. Martyn Cundy in private correspondence decades ago.  He was not impressed.  However, I'm sure it's at least as good as Euclid's Formula; upon which it was based.  So I not claim a major breakthrough in Number Theory.  That is to say, it did not impress a famous mathematician (i.e. of "Cundy & Rollett"), but is better than the stuff currently universally offered at the high school level.  [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain|talk]]) 16:30, 13 January 2022 (UTC) 
:Thanks for your comments. However, Wikipedia policy is that [[WP:OR|original research]] is strictly forbidden in Wikipedia, and, on such a topic, everything must have been published in  textbooks. So, your personal notes are useless here. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:45, 13 January 2022 (UTC)
::There is a whole article devoted to [[Formulas for generating Pythagorean triples]].  Perhaps the approach you are looking for is listed there?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 23:53, 13 January 2022 (UTC)
"Drats. Spoiled Again."  I refer to the comments appended by D. (Daniel?) Lazard and "Quantling".  The first informs that original research is not allowed on Wiki-pedia.  However, when I previously tried to explain my work on "Axonometric Projections" in that topic's 'Article' section, it was removed for the same reason; even though my printed article had been published in 1989 in EDGJ.  (See my area in that topic's 'Talk' section; where I have at least been able to cite the recent Digital Object Identifier for my 'Correction' to that original EDGJ article.  Since I'm the author of the printed article, it seems that I have no source to cite.)

Citizen Quantling suggests I look around.  ('Surely someone has anticipated you?')   I've not tried to publish my version of parametric expressions for A, B, and C for over 30 years; during which time, another source has not come to light.  To be sure of this, the other day I quickly reviewed all the sites I could find, and found them all basically repeating Euclid's approach.  Mine, of course, is equivalent, but no one seems to have taken the same approach.    [Well, the ancient Greeks were all over such geometric problems, and may have been way ahead of me, but we only have Euclid's as the surviving method.]  I hope he or she was not meaning to be condescending.  Thank you, but Prof. Sloane told me years ago that it is not easy to find anything new in mathematics.  Mine is not a new discovery, but only a reworking to answer a quibble I had with the Euclid presentation. 

So, this frees me up today to do some reading instead. <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain#top|talk]] • [[Special:Contributions/Lemchastain|contribs]]) </span>
:Please read [[WP:Original research]]. Publishing your own research in Wikipedia is strictly forbidden per [[WP:OR]], and also [[WP:COI]]. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 19:03, 15 January 2022 (UTC)

I just added Sierpinski's "Pythagorean Triangles" to the references section of the article.  Apparently, you don't add 4 tildes at the end of a reference.  Now I'm wondering how long it will be left up.  It was first published in Polish in 1954; then Russian in 1959; and then English in '62.  It was Number Nine in 'The Scripta Mathematica Studies' series.  ([[User talk:Lemchastain|talk]]) Ljc 00:33, 8 February 2022 (UTC)
:I removed it again, because it was already in the footnotes (in the Dover edition) and your addition made a duplicate reference. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 00:47, 8 February 2022 (UTC)
:: {{Reply to|David Eppstein}}  This does not surprise me -- although it took you almost a quarter of an hour.  However, I would like to raise a couple of points:  1. You use the word 'again'.  I do not recall trying to list Sierpinski prior to this.  Others may find in the records that I did; or perhaps someone else did.  2.  You speak of "footnotes".  I do not see that term listed in the articles index.  I have searched the references, and no Sierpinski.  In "Notes", #13 does list his name, but only pp. 4-6.   I was trying to list his entire book.  How would you like a 'reference' to Sierpinski(1962), pp. 1-3, and 7-107?   Surely, someone interested enough in this topic to start going through the 'references', 'notes', and 'See also' might well want to know what was written in 1954, and which to a good extent is just being repeated without citation on several Internet sites.  To carry this a bit further, since Euclid is cited (in the Heath translation), why not just 'revert' most of the other repetitive citations.   Also, note 13, does not say explicitly what bit of Sierpinski's writing to which it refers.  He must have written at least 4 other pages on something else at some point.  If anything should be 'reverted' it is such a vague citation as note 13. ([[User talk:Lemchastain|talk]]) Ljc 16:57, 8 February 2022 (UTC)
::To say nothing (almost) of the difficulty in finding a reference in the 'Notes' section; due to it not being in an alphabetical order and being a bit long.<!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain#top|talk]] • [[Special:Contributions/Lemchastain|contribs]]) 15:29, 8 February 2022 (UTC)</span>

== Trigonometric functions of the angles are rational ==
At their request, I am looking for a citation to back the [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&type=revision&diff=1065853887&oldid=1065853501 property just reverted] by {{reply to|DVdm}}.  Anyone have one?  The text is
<blockquote>
*Each of the acute angles of a Pythagorean triangle has a positive [[rational number|rational]] sine and cosine.  That the sine is rational and positive follows from the fact that the area and sides are positive integers and area formulas like {{math|1=''Area'' = (''ac'' sin ''β'') / 2}}.  That the cosine is rational and positive follows from the [[law of cosines]] formulas like {{math|1=''b''{{sup|2}} = ''a''{{sup|2}} + ''c''{{sup|2}} - 2''ac'' cos(''β'')}}.  The tangents of the half angles are positive and rational as well because {{math|1=tan ''β''/2 = (sin ''β'') / (1 + cos ''β'')}}.  The reverse is also true: any rational number in the interval {{math|(0, 1)}} can be used to construct a primitive Pythagorean triple with that number as the half-angle tangent of one of the acute angles; see [[formulas for generating Pythagorean triples]].
</blockquote>
Alternatively, do we have consensus that this is [[WP:CALC]]?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 17:25, 15 January 2022 (UTC)
:Let's find a source first, as this is not just CALC's ''routine calculations'' or  ''basic arithmetic''. Happy source hunting! - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 17:36, 15 January 2022 (UTC)
::[[WP:CALC]] includes 'Mathematical literacy may be necessary to follow a "routine" calculation, particularly for articles on mathematics or in the hard sciences.' and I find the above text to be reasonable because it does not demand much mathematical literacy.  But it is not up to me alone!  I respect {{reply to|DVdm}} and their opposition to [[WP:CALC]] in this case means we need a citation.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 17:51, 15 January 2022 (UTC)
:::{{reply to|DVdm}} if only part of the text is beyond [[WP:CALC]] by your evaluation, perhaps the remainder can be re-inserted.  Thoughts?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 17:56, 15 January 2022 (UTC)
:::: The substantive content of this addition is already covered elsewhere in the article -- see [[Pythagorean triple#Interpretation of parameters in Euclid's formula]] and [[Pythagorean triple#Rational points on a unit circle]].  And the presentation choices of this addition are questionable.  --[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 18:25, 15 January 2022 (UTC)
:::::Pardon me for intruding, but I could not help noticing your question when I was proof-reading my section just above.  So, you need to cite a source for that?  "Rediculous!", if you will allow me some theatrics.
:::::Just cite Euclid; or just about any trig. book since.  Having uttered the words, "Pythagorean Triangle" you should not even have to mention that 3 of the trig. functions are rational.  Also, on the Euclidean plane, given a right triangle with a rational tangent less than one, you are guaranteed a Pythagorean Triplet since you are doubling the smaller acute angle.  Cite the double-angle formulae from "Trigonometry - Plane and Spherical", by Miles C. Hartley, c. 1942 (sic!  I gave away my good trig. book by accident, but still have this $1.00 purchase.)   You will see from above that I am having my own problems with sources.   However, my Talk section has not been deleted; so I may just post everything there, and not have to worry about endless edits of my text.[[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain|talk]]) 18:27, 15 January 2022 (UTC)
:::::: {{ec}} {{rto|Lemchastain}} one of the reasons why Wikipedia requires sources, is related to notability: some statements or claims can be 100% true and correct, but if nobody has ever mentioned them in a relevant article or textbook, they have no place in Wikipedia—by sheer design. Regarding your talk page, the last version of it that made sense to me is [https://en.wikipedia.org/w/index.php?title=User_talk:Lemchastain&oldid=1009557406 this one]. What you [https://en.wikipedia.org/w/index.php?title=User_talk%3ALemchastain&type=revision&diff=1010125001&oldid=1009557406 did with it], resulting in [https://en.wikipedia.org/w/index.php?title=User_talk:Lemchastain&oldid=1010125001 this], is not parsable for me. - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 19:03, 15 January 2022 (UTC)
:::::::{{ec}}The given proof is ridiculous and does not belong to an encyclopedia. I have no opinion whether the result is worth to be included here, but, if included, it must be with a formulation such as {{tqq|The values of the [[trigonometric functions]] of the acute angles of a Pythagorean triangle are positive [[rational number]]s, since these values are ratios of integer lengths of sides of the triangle.}} As such, [[WP:CALC]] applies to the proof, but sources are required for the [[WP:notability]] of the result. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:50, 15 January 2022 (UTC)
::::::::{{reply to|D.Lazard}} Right you are when you say "The given proof is ridiculous" and I am embarrassed; it's a good thing that I am not the only Wikipedia editor.  I like your text.  Personally, I would support it without a citation, but because I am not the only Wikipedia editor, I am hoping we can find a citation to support its [[WP:notability|notability]].  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 01:37, 17 January 2022 (UTC)
:::::::::17 Jan 2022, Lem -- Dear Quantling,  I had to get up to send you a note on my comment just above.  You did not seem to know anything about it being 'hidden'.   Now it is not hidden again.  May I assume that you had the last word on that, for now?  I had not checked the 'article' before writing "'Rediculous!'".  I only meant it did not seem necessary to give a source on well established planar trigonometry.  If, however, someone was instead saying that your discussion was not needed because it only repeated well known material, then, I spoke out of turn.  It seems either it is not needed, but if allowed, does not need a source.  After you read this, feel free to delete it.  {Apparently, nothing is ever really deleted anyway(?):  do the edit police keep a record of deleted material?   
:::::::::Now, as to the immediately following entry:  I've read it, and would not care to see it again; especially everything from "Regarding your talk page, the last version..." on.  My talk page has nothing to do with this topic.  Also, as I deleted some parts that were not worth the space taken, I don't want others to waste time on something that is no longer there.  If I recall correctly, a day or two ago, I suggested to "DVdm" that I was willing to delete my entire page (to end the quibble).  It is supposedly _my_ talk page after all.
:::::::::========== [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain|talk]]) 04:22, 18 January 2022 (UTC)
:::::::::[[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain|talk]]) 03:55, 18 January 2022 (UTC)
::::::::::Thank you {{rto|Lemchastain}} for your ongoing contributions to this page and for your support.  For the present matter, I think that we have consensus that the addition to the article included an overly complicated proof.  The proof can easily be simplified to the point that it is considered trivially true under the [[WP:CALC]] guidelines.  However, an editor has requested that we find a citation; not to prove the truth of the text but to show that it is notable enough to be worth of inclusion.  This is a reasonable request, so there will be a slight delay as this is sought out.  Thanks again for your support —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 23:06, 18 January 2022 (UTC)

== very minor suggestion ==

Under "Euclid's formula":
An early sentence has "and not both odd" (referring to m and n).
The next sentence has "one of which is even" (referring to same).
I would think it better to use the latter expression in both sentences.
[[User:Dcromley|Dave C]] ([[User talk:Dcromley|talk]]) 16:50, 11 March 2022 (UTC)

:I agree, it’s simpler.  I’ve made the change. —[[User:JayBeeEll|JBL]] ([[User_talk:JayBeeEll|talk]]) 17:16, 11 March 2022 (UTC)

== Recent work by A. Yekutieli (me) ==

Hello Editors.

I made a few attempts yesterday to add this new material; before there was an attempt by Liran Shaul. All was deleted, immediately. 

I propose the editors involved in this entry take a look at the material, esp. the arxiv eprint https://arxiv.org/abs/2101.12166. 

My paper was accepted by Amer Math Monthly, to appear within the next few months. The published paper will be similar in content to the epirnt.  

Let me add a few more things.
# In general I enjoy using English Wikipedia as a reference form many things. From math (I am a math professor) to trivia. I also contribute money to the foundation.
# The material I proposed to add is important, not only in my biased opinion, but also in the eyes of the referees and editors of Amer Math Monthly, where my paper was accepted for publication.
# I suggest that one of the regular editors who handle the entry "pythagorean triples" will take a look and enter my contribution once they see it fits in. I added some names in the cc.
# I will refrain from tying to edit wiki entries in the future. I have no patience nor time for this very regimed practice. Yesterday's efforts were the first and last for me.

All the best,
Amnon Yekutieli
@MrOllie @User:D.Lazard @User:Anita5192 @User:David Eppstein Amyekut (talk) 07:58, 21 June 2022 (UTC)  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Amyekut|Amyekut]] ([[User talk:Amyekut#top|talk]] • [[Special:Contributions/Amyekut|contribs]]) </span> <!--Autosigned by SineBot-->
:{{reply to| Amyekut}} I made an edit.  Let me know what you think.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 23:39, 23 June 2022 (UTC)

::Dear @[[User:Quantling|Quantling]],
::A mention in "Notes":
::[25] Yekutieli, Amnon (8 Jan 2022). "Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers". arXiv:2101.12166 [math.NT].
::is better than nothing, of course. 
::But it would be much more useful and informative if a paragraph summarizing my new ideas is added. It could something like what I tried to do a few days ago, or something you -- or another editor -- writes, based on my eprint and the lecture notes. 
::The version that will appear in Amer Math Monthly is much better than the eprint (all proofs are written, and the expostion is improved), but the paper will only be published in 2023, and even then it will be under embargo for a year. 
::S[https://drive.google.com/file/d/1akuakZSG3Db-JGLW2a735XJvynbmrWyu/view?usp=sharing ee this pdf file for all related information.] (This link should not be made public.)
::Thanks,  Amnon  [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 10:01, 24 June 2022 (UTC)
:::As to the derivation and underlying math, I am hesitant to discuss those in the Wikipedia article.  (That doesn't mean that you shouldn't keep advocating for that yourself, only that I am not a likely part of that process.)  If there is another stand-alone result from the arXiv paper that you think would also make a good note, what is it?  I could add that as a note too.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 14:44, 24 June 2022 (UTC)
::::Has this been published yet? We shouldn't be citing preprints on Wikipedia. And we generally shouldn't be writing about new results until there are secondary sources available. [[User:MrOllie|MrOllie]] ([[User talk:MrOllie|talk]]) 14:58, 24 June 2022 (UTC)
:::::You can wait until my paper will appear in Amer Math Monthly. The editor told me it would be some time in 2023. 
:::::Since the ideas are 3,000 years old, what's another year? 
:::::However, we mortals have little patience and also finite life spans. So you might consider something, see suggestions in my  [https://drive.google.com/file/d/1akuakZSG3Db-JGLW2a735XJvynbmrWyu/view?usp=sharing other pdf file]
:::::Here is my  [https://drive.google.com/file/d/11QUbCqnAU7tvc6gfumqk6-Ly7Zn8cgdC/view?usp=sharing main theorem and some examples.] [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 17:50, 27 June 2022 (UTC)

::::::Hello.

::::::I am new to the Wikipedia territory, and of course I am totally biased, since this is about my own work. 

::::::I noticed that @Quantling has added a reference to my work (the eprint, and the paragraph 

:::::::For every prime number ''p'' for which {{nowrap|1=''p'' mod 4 = 1}} and for each positive integer ''n'' the hypotenuse {{nowrap|1=''c'' = ''p{{sup|n}}''}} occurs in exactly one primitive Pythagorean triple with {{nowrap|''a'' < ''b'' < ''c''}}.  More generally, if {{nowrap|1=''c'' = ''p{{sub|1}}{{sup|n{{sub|1}}}}'' ... ''p{{sub|k}}{{sup|n{{sub|k}}}}''}} is the product of ''k'' such values (with distinct values ''p{{sub|1}}'', ..., ''p{{sub|k}}'') then there are exactly 2{{sup|''k'' − 1}} such primitive Pythagorean triples.  No other values of ''c'' are the hypotenuse of a primitive Pythagorean triple.<ref>{{cite arXiv |last=Yekutieli |first=Amnon |author-link= |eprint=2101.12166 |title=Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers |class=math.NT |date=8 Jan 2022 }}</ref>

::::::This is great!

::::::However:
::::::# The number 2^{k-1} refers to '''normalized PTs'''. For each normalized PT (see my eprint) there are '''two primitive PTs''', so their number is 2^k. (I have read that this was dicovered by C.F. Gauss, but I did not see any reference to his work on this. It is a historical mystery.)
::::::# My work not only counts, but also '''presents all primitive PTs with given hypotenuse c'''. This appears to be a '''genuinely new discovery'''. 

::::::Thus I believe you -- the dedicated expert editors -- should add more of my work to this entry. My eprint does not have all full proofs, and has less examples, than my accepted version of the paper. The latter has restricted access only now, but if any paticular editor wants to see a copy of it, without distribution, I can try to clear this with the editor of AMM. 

::::::Thanks, Amnon  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Amyekut|Amyekut]] ([[User talk:Amyekut#top|talk]] • [[Special:Contributions/Amyekut|contribs]]) 10:42, 25 June 2022 (UTC)</span> <!--Autosigned by SineBot-->
:::::::To: Editors at Wikipedia
:::::::Regarding the entry
:::::::Pythagorean triple
:::::::https://en.wikipedia.org/wiki/Pythagorean_triple
:::::::Maybe the time has come for you to add a full discussion of my work in the paper
:::::::Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
:::::::The American Mathematical Monthly Volume 130, 2023 - Issue 4
:::::::DOI https://doi.org/10.1080/00029890.2023.2176114
:::::::As independent reference, please see the papers
:::::::[JMMM1]    Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples
:::::::Thomas Jaklitsch, Thomas C. Martinez, Steven J. Miller, Sagnik Mukherjee
:::::::https://arxiv.org/abs/2112.03663
:::::::and
:::::::[JMMM2] Class Numbers and Pell's Equation x2+105y2=z2
:::::::Thomas Jaklitsch, Thomas C. Martinez, Steven J. Miller, Sagnik Mukherjee
:::::::https://arxiv.org/abs/2203.16709
:::::::These papers were inspired by my ideas. Indeed, in the abstract to [JMMM1] they say
:::::::"Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves; for the first, we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently Yekutieli discussed a connection between these two problems and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. We generalize these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given z to x2+Dy2=z2 when D is 1 or 2 modulo 4 and the class group of ℚ[−D‾‾‾‾√] is a free ℤ2 module, which always happens if the class number is at most 2. We give examples of when the results hold for a class number greater than 2, as well as an example with different behavior when the class group does not have this structure."
:::::::Yours,
:::::::Amnon Yekutieli [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 12:50, 10 March 2025 (UTC)
::::::::According to Google Scholar; [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C10&q=Pythagorean+Triples%2C+Complex+Numbers%2C+Abelian+Groups+and+Prime+Numbers&btnG= Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers] has been cited once; [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C10&q=Connections+of+Class+Numbers+to+the+Group+Structure+of+Generalized+Pythagorean+Triples&btnG= Connections of Class Numbers to the Group Structure of Generalized Pythagorean Triples] has been cited three times; and [https://scholar.google.com/scholar?hl=en&as_sdt=0%2C10&q=Class+Numbers+and+Pell%27s+Equation+x2%2B105y2%3Dz2&btnG= Class Numbers and Pell's Equation {{math|1=''x''{{sup|2}} + 105''y''{{sup|2}} = ''z''{{sup|2}}}}] has been cited once.  Given the thousands of years of history of Pythagorean triples, with at least thousands of papers having at least hundreds of citations each, I expect that it will be impossible to show that these three papers are sufficiently noteworthy for this encyclopedic article. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:03, 10 March 2025 (UTC)

:::::::Thanks [[User:Amyekut|Amyekut]], you're right to stay clear of [[WP:SELFCITE]] issues.
:::::::Pinging [[User:Quantling|Quantling]] who added the text. [[User:Cabayi|Cabayi]] ([[User talk:Cabayi|talk]]) 10:55, 25 June 2022 (UTC)
::::::::I see now that @Quantling stated the correct fact re enumeration, demanding that a < b < c. 
::::::::I use the new name "normalized" for such primitive PTs. 
::::::::From my POV, there is little use for the traditional attribute "primitive", since what we are interested in really is PTs up to similarity of triangles (the action of the group O_2(R) \ltimes R^2, not only the subgroup SO_2(R) \ltimes R^2).  
::::::::AY  [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 11:30, 25 June 2022 (UTC)
:::::::::Hello all interested parties: I am making an edit to describe how to enumerate the triples.  If you think it needs changes and if you have the time, your editing it rather than wholesale reverting it would be much appreciated.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 21:32, 27 June 2022 (UTC)
::::::::::I rewrote the paragraph for separating assertions from proofs. and also for clarification and less technicalities (avoiding "mod"). The tag {{tl|citation needed}} referred to a basic result of the theory of [[Gaussian integer]]. So I started to replace it with a link to [[Gaussian integer]]. But I remarked that all the proof is a corollary of the theory of Gaussian integers, and that Gauss introduced it for solving this sort of problem (maybe, exactly this one). So, I have added a proof that is probably in Gauss' work (I have not checked). I have kept the reference to Yekutieli, but I am not sure it is worth to keep it (to be discussed).
::::::::::By the way I have removed the mention to a procedure "to enumerate", as only a way "to count" is provided. An enumerating procedure should include a procedure to express a prime as a sum two of squares. I do not know whether there is such a procedure that is better than exhaustive search. Even if there is such a better procedure, I am not sure that this would give something more efficient than the exhaustive search for decomposing ''c'' into sums of two squares, since the direct decomposition of ''c'' avoids a possibly costly factorization, and also avoids to repeat the procedure of decomposition into sums of squares. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:59, 28 June 2022 (UTC)
:::::::::::Thank you for the edit.  Yes, I too was worried about how easy it would be to find the two squares that add up to the prime.  One way to do it is to start with the primitive Pythagorean triple that has its hypotenuse equal to that prime, but that doesn't help!  A way this recipe does help with enumeration is that it tells us that we need to do the hard part only once for each prime; not every value of ''c'' independently.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 22:22, 28 June 2022 (UTC)
::::::::::::To be clear, '''I''' do not know whether there are efficient algorithms for this problem. But, viewing the amount of work done in computational number theory, I am quite sure that the problem has been studied, and either efficient algorithms have been found, or it has been proved that there cannot be algorithms that are significantly better than exhaustive search. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 08:34, 29 June 2022 (UTC)
:::::::::::::to @[[User:D.Lazard|D.Lazard]] 
:::::::::::::1) I am glad that a piece of my work was included, as a digest. 
:::::::::::::2) I do not think there is any written or published proof of the counting of normalized PTs, besides mine. I mean the formula 2^{k-1}. I did a search, and so did the anonymous referees of my paper at AMM, and we got nothing. The closest is an oblique mention of a "Gauss" in  a paper from around 1890. (C.F. Gauss was long dead by then.)  
:::::::::::::3) I do not know any way to find the prime decompostion of 5, 13, etc in the ring of Gauss integers, besides the sum of two squares. 
:::::::::::::4) Regardless of 3), there is what in my mind a really nice fact, which is to quickly write down all the normalized PTs with hypotenuse a power of p. This is quite satisfying even for p=5. See this note 
:::::::::::::[https://drive.google.com/file/d/1qunvuQ7aF6ZbupOG69nQQ98fSGD00Qaz/view?usp=sharing] [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 20:12, 8 July 2022 (UTC)
::::::::::::::1)Nothing of your work is included, except the result, which is certainly not new (see below). The proof is not yours. I kept the citation for attesting that the result is not an original reseach of Wikipedia editors.
::::::::::::::2) "I did a search": For this kind of question, the fact that you did not succeed does not mean nothing because (a) not all old works have been numerized (b) it is difficult to find such a result by key-word search, because it may not appear in any article title, and it may be formulated in many different ways. 
::::::::::::::"I do not think there is any written or published proof ...": My experience is that a question has ''always'' been considered before if it is worth to be considered, and if it belongs to an area that is widely studied from centuries. If the answer to the question can be found in a few minutes by any specialist of the subject (I am not a specialist, and I got the proof in a few minutes), it is sure that the proof has been written somewhere. The difficukt problem is to find it, as, in this case, it could appear as an exercise left to the reader, or in an application or example treated in few lines in a book or a long article. So, I am pretty sure that the result is not new. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 11:32, 9 July 2022 (UTC)
:::::::::::::::@Lazard: I will not reply directly to your argumentative and insulting (and boorish) remark. It is sad. I will reply obliquely. 
:::::::::::::::'''Experts on this subject, and also the anonymous referees for AMM, were not aware of this fact (my proof and construction) before I wrote my paper.''' Neither were you. 
:::::::::::::::I am quite certain you won't find my lovely '''table of powers of 5''' anywhere (prior to my work). 
:::::::::::::::The fact that you can understand my arguments is fine -- my paper at AMM is intended for undergraduate students. So there you are. 
:::::::::::::::'''But you did not think of it on your own, did you? Neither did anybody else''' (the onus is on you to prove otherwise). 
:::::::::::::::'''Discovery is not always hard to understand.''' 
:::::::::::::::Are you be any chance jealous of me? If so then you should probably refrain from editing this item...  [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 18:12, 10 July 2022 (UTC)

::::::::::::::::I do not follow the part of the article that currently reads
:::::::::::::::::Now, the [[unique factorization domain|unique factorization property]] of Gaussian integers implies that, if the prime number ''p'' is the norm of {{nowrap|''a'' + ''ib''}} with {{nowrap|''a'' > 0}} then it is also the norm of {{nowrap|''a'' – ''ib''}}, but it is not the norm of any other Gaussian integer with positive real part. The unique factorization property thus gives {{nowrap|2{{sup|''k'' − 1}}}} for the number of primitive Pythagorean triples with hypotenuse ''c'', taking into account that the change of sign of all imaginary parts gives the same triple.
::::::::::::::::The proof in the cited work instead works with {{nowrap|''m{{sub|j}}'' ± ''in{{sub|j}}''}} where the norm is {{sqrt|''p{{sub|j}}''}} rather than ''p{{sub|j}}''.  In the cited work it is the choice of signs in these expressions that leads to the count {{nowrap|2{{sup|''k'' − 1}}}}.  Should I edit the article accordingly or am I missing something?  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 19:56, 10 July 2022 (UTC)
:::::::::::::::::To @Lazard: Your preposterous claims are really annoying me. I think more assertive words are in place. 
:::::::::::::::::I claim that in stating this proof publicly, but without attribution to me, constitutes '''plagiarism''' on your part. 
:::::::::::::::::You have read my work, and then reproduced a derivative (and possibly faulty) version of it, without proper attribution. 
:::::::::::::::::I will let you contemplate, search, consult, etc. for two weeks from now. If by then you are unable to show a reference to this proof and construction that predates mine (this means earlier than 2010, I think), and yet you won’t give me proper credit, I shall take the necessary measures. 
:::::::::::::::::I am signing off from this discussion, and placing a reminder in my calendar. In the mean time I can be reached by email at  amyekut@gmail.com.  [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 21:24, 10 July 2022 (UTC)

I have restated the results in a less technical way. In particular, I have removed the explicit prime factorization. For not polluting the list of properties, I have moved the proof in a footnote and, hopefully, clarified it. {{to|Quantling}} In the proof, this is the Gaussian norm rather than the absolute value that is used, the former being the square of the latter. Among other advantages, this allows not considering [[irrational number]]s (square roots). [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:11, 13 July 2022 (UTC)
:{{reply to|D.Lazard}} thank you for the clarification and for your recent edits.  —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 20:21, 13 July 2022 (UTC)

::Has two weeks since 10 July 2022 passed yet?   I have very recently added the next topic below, but find this broahaha irresistable.  The only problem is catching up on all the comments that were flying about.

::First of all, Amnon, I understand your initial frustration:  see my similar feelings in the topic "Lemchastain comments" above in this talk section.   However, I tried to contact you about being an 'ArXiv endorser' some time ago (at ...bgu.ac... address) and you did not reply -- to decline, or otherwise.  (It is a moot point now due to the Overmars and Ntogramatzidis paper on arXiv that I reference below.)   I just took a quick look at your paper on arXiv.  It is the 21st to come up "Pythagorean triple", with L. and N. being the 53rd.  I suggest you look at their paper.  You might also look at mine:  I sent you a copy on May 18th.  (That was not the final version.  Would you like a copy of that final version?  I have 2 E-mail addresses for you.)

::If I found the L. & N. overly complicated for generating Pythagorean triples, yours was over my head.  I will leave it to others to judge its merit.  I do reject the idea that separating the primitive triples from non-primitive Pythagorean triples is not important -- if that is what you intended to imply somewhere above.

::O. & N. got the ideas of using series and of changing the (old) 'm' to a smaller parameter; which they in a stroke of capricious genius called "m'!!!  So, their new (m,n) is not the old (m,n).  [I'm a simple man, so that is confusing to me.  Others may insist that their's has a different emphasis -- or accent?]  In any case, I'm with them mostly.  You will see in my Part V (June 24th version) I do mention the possibility of imaginary parameters to allow 'c' to be negative.  I could not figure out if the imaginary values you show in your paper were at all related.  If you read Part V, let me know.

::Now to a big question.  I tried to put something about my idea on Wiki because it looked like it would be a quick and easy way to make it public.  You know as well as I how deceiving looks can be.  But you are going to be published in a recognized mathematics journal, and after peer review; so WHY are you bothering with Wiki now.  After someone reads the published article, let them struggle with the edit police to summarize it on Wiki.

::I have an observation to make on one other arXiv article, but think I should put that in the O. & N. section.  (I expect to let my eponymous section above wither on the vine; since it has been dormant for months.)  I hope you get this.  In the time that has passed, I have forgotten how to send copies to people.  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain#top|talk]] • [[Special:Contributions/Lemchastain|contribs]]) 20:40, 23 August 2022 (UTC)</span> <!--Autosigned by SineBot-->

{{talk ref}}

=== New paper on this subject ===

Title: Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Author: Amnon Yekutieli

Ref: 

''American Mathematical Monthly'', online open access, 
<ref>https://doi.org/10.1080/00029890.2023.2176114</ref>

[[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 10:46, 11 March 2023 (UTC)

{{Reflist-talk}}<!-- Keep this at the end of this section. -->

== 2015 paper by Anthony Overmars and Lorenzo Ntogramatzidis (and similar in 2018 w. Venkatraman)  ==
:Some months back, I tried to express the view that the 'm' and 'n' parametric equations were obsolete.  Since I was basing this on my own work, I got no where with the wiki-pedia topic.  I can now point to a published paper by others that should support my earlier view.  Note, this time, I'm recognizing two Australian professors.
:To put it simply, I've been scooped by several years by these authors.  I discovered their 2015 paper after the Chair of a mathematics department told me that I needed to list references in the paper I had considered finished (after a couple of months of revisions).  When I saw the title. "A new parmeterisation of Pythagorean triples in terms of odd and even series", I feared the worst; which was there.
:I had my 3 parametric equations for sides, 'a', 'b', and 'c' from the mid-80's.  I set them aside when Prof. H. Martyn Cundy said they weren't important.  Since those equations were a one page affair, I added material about the use of series to simplify computations for all three sides.  The Overmars and Ntogramatzidis paper covered both of those things.  One of their charts was very much like one of mine.  [The only thing I had left was the idea of allowing for negative values for 'a' and 'b' -- and 'c' if imaginary numbers were used.  However, while these satisfy a^2 + b^2 = c^2, they are not in keeping with the geometric problem.]
:I did all of the work on the paper before finding out about either of their two papers.  I have tried to E-mail the authors, but have had no reply thus far.  Since I have sent out a lot of copies of my paper, I wanted them to know that I'd only just come across their __prior__ paper.
:A minor warning is in order though.  Their notation may be somewhat confusing, due to unfortunate choices for various parameters.  Instead of starting with the usual parameters 'm' and 'n', they start with 'u' and 'v', respectively.  Eventually, they conclude that 'v' = 'n'.  This is the same 'n' as used in the usual expressions, and the same I used.  That is, there is no need to change 'n' at all.  For the usual 'm' -- this where it may be very confusing -- they come up with (n + [2m-1]).  Their 'm' is not the one normally seen!   Where they have [2m-1] may paper uses [2t+1].   This is not a significant difference:  both are making sure this quantity is uneven, but the re-use of 'm' is not helpful.  Since neither my 't' nor their new 'm' is the real parameter, I used 'u' = [2t+1] to emphasize that my new parameters are a new 'u' and the original 'n'.   [I didn't know they had used 'u' for 'm' --if you can follow this.]  
:On page 5, they give d = (2m-1)^2.  So d is the same as my u^2.  
:On their page 4, 'a' = (n + 2m-1)^2 - n^2; 'b' = 2(n = 2m-1)*n; and 'c' = (n = 2m-1)^2 +n^2.  {Hopefully, with more clarity, my equivalent equations are: 'a' = u(u + 2n); 'b' = 2n(u+ n); and 'c' = (u + n)^2 + n^2.
:To keep from going crazy, one might distinguish their m with an accent or subscript.
:They have a lot more notations, and symbols not meant for a non-professional.  (Lacking their training, I had to avoid fancy symbols.)  
:In conclusion, although a bit obscure, the Overmars and Ntogramatzidis 2015 paper does make the usual 'm' and 'n' parametric equations obsolete.  One just has to struggle through some notational development that may not be important to the average user.  [I have not sent out any copies of my paper since they pulled the carpet out from under me.  To the extent that I have not gotten replies, I have not sent out a 'mea culpa'.  If I get any inquiries in the future, I will have to cite A.O. and L.N.  With due humility, my paper would at least help someone figure out what they are getting at -- I think.] ([[User talk:Lemchastain|talk]]) Ljc 19:58, 22 August 2022 (UTC)

::What is the citation of the source?—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 20:44, 22 August 2022 (UTC)

:::Oh, I must begin by thanking you replying to the above.  (I think you have made the topic permanent.)  I'm sorry about not giving the reference.  I meant to include it, but the comment got a bit long and I forgot.  Also, I wasn't sure the new topic would be allowed to stay up this long (i.e. several hours so far).  Finally, there was something screwy going on with the format:  notice three sections  -- which were not done on purpose.
:::But to get to your question, and take a much bigger chance of being erased.  The short answer is arXiv.org, but I will elaborate after an aside. 
:::When I gave up on trying to get any of my work on wiki-pedia, someone suggested that I try "arXiv"  It is handled by Cornell U. so seemed legit.  (Actually, it is huge, e.g. about 92 professional articles appear for the topic "Pythagorean triple" -- or, "P. triples"; can't recall whether singular of plural.  As a disadvantage, the papers are not peer-reviewed, but it does accept simple .pdf's.  When I considered trying to write for other publications, they had strict guidelines which were stumbling blocks for me.  One wanted LaTex. I don't know if or have it.  Another insisted on Word.  Ditto.  I had the LibreOffice suite, but didn't know it either.  From the "Writer Guide" and the "Draw Guide", I learned just enough to complete my paper, informal as it was.  My nephew lambasted its short-comings:  no abstract, etc.  Eventually, I learned I just had to add references.  So, I started my search of the papers in arXiv.  They were all over the place.  Many I would not try to read, but was able to reject as at all relevant, until I reached the 53 article on Pythagorean triples.  That is the O. & N. one from 2015.  (I'm getting there, Anita.  I hope you can see this when I'm done.)
:::The main one of interest is "arXiv: 1504.03163v1 {math.Ho] 1 Apr 2015", or "A new parameterisation of Pythagorean triples in terms of odd and even series", by Anthony Overmars and Lorenzo Ntogramatzidis, _then_ of Curtin University, Perth (WA), Australia.  Overmars has moved on, but I can't figure out where.  L. N. may still be there but has not responded yet.
:::The same nephew alerted me to a second paper from 2018 that appears on aimspress.com(period)
:::That one adds a third author, Sitalakshmi Venkatraman.  At that time she and Overmars were at Melbourne Polytechnic.  I failed to reach her at the E-mail address given.
:::This 'Research article' is called "A new approach to generate all Pythagorean triples", AIMS Mathematics, 4(2): 242-253 (period)  Also know as DOI:10.3934/math.2019.2.242; with 242 being the first page of the article (in that issue?).
:::If the 2015 article was somewhat obscure, this one gets even worse.  The first part, seems to just recapitulate the 2015 article.  Of interest to me was their Figure 1.  It is similar to my Diagram 1, but they fail to recognize the importance of the inscribed circle.   I do not know how to get a copy of my "paper" to you; unless you can figure out how to E-mail me.  [I once tried to put my E-mail address on wiki, before I was warned against that.  If it survived it is either in Pythagorean triple of in Axonometric Projections, but then it may not interest anyone here anyway; especially those able to wade through O. & N. or O, N, & V.]
:::Aside from the revised parameters that supplant the old 'm' (not their 'm'!), the use of series that they also cover has been neglected in the literature I've seen.  [I'd scrap just about everything that comes up for an Internet search:  old 'm' and 'n' everywhere.]
:::Thank you for your patience, Lem ([[User talk:Lemchastain|talk]]) Ljc 02:21, 23 August 2022 (UTC)
::::As to the Pythagorean triple topic an "arXiv" (X pronounced as 'chi', I think), one paper should give us caution.
::::The 14th article, a bit above Yekutieli's in 21st position, is by Kyle Bradford.  I don't make these things up:  the title is, "Pythagorean Triples, Bezout Coefficients and the Erdos-Strauss Conjecture".  Proceed at your own risk.  Really, I tried to write an article that a smart high school student could understand.  (The hardest term is probably "series"; which I looked up, so as to make sure I didn't mean "sequences".)   It does not surprise me that Bradford and also Yekutieli have gone unnoticed by posting on arXiv.  The papers are not peer reviewed, and sit there in hopes of being picked up by a recognized journal.  (Kudos to Amnon for attaining that goal.)
::::Bradford is another story, and an instructive one.  Although I was pretty much put off by the title, I did turn a few pages and came across something I felt I understood.   His Figure 1 is on page 4.  It lists 27 triples.  Below the image, it says, "This image of a Breggren Tree is thanks to Louis Teia. [4]"  Since I have a table of many triples, I happened to find an error in the listings.   While Mr. Teia should not have let it slip by, the author can't blame Louis:  he should check anything he is putting in his paper.  Secondly, this shows a problem when lacking peer-review:  a reviewer or editor would have checked all the entries.  After finding one error, I did check the others.  I found two more.  The most egregious was the 20th from the top: [252, 276, 373].  If you know this topic at all, that should jump out at you; even more so if you are writing about it.  (You are welcome to look for the other two errors.)  Thirdly, this shows the advantage of deriving triples by using series, as O. & N. do.  That was also a big part of my paper; so they got me on the two main things I wanted to proclaim to the world! [Drats!]  I assume that Louis Teia did not use series to find triples.  The 'Breggren Tree' does not seem to work in the same way as a two-dimensional table of triples.  Was he then reduced to finding triples through more demanding computations?  
::::I mentioned the errors to arXiv.  they wrote back that only authors may change a paper.  I tried to reach to the E-mail address that arXiv had:  no luck.  I then tried him at a college in Georgia, to which he had moved  again no response, but he may have moved again.   [The listed E-mails on arXiv are often out of date.]  <!-- Template:Unsigned --><span class="autosigned" style="font-size:85%;">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Lemchastain|Lemchastain]] ([[User talk:Lemchastain#top|talk]] • [[Special:Contributions/Lemchastain|contribs]]) 21:24, 23 August 2022 (UTC)</span> <!--Autosigned by SineBot-->

Dear SineBot, Is thius better? ([[User talk:Lemchastain|talk]]) Ljc 18:09, 26 August 2022 (UTC)

:Update:  In which he talks to himself, and hopefully some others -- in addition to the 'editors'.  
:I did get one E-mail from L. Ntogramatzidis.  He confirmed that A. Overmars has moved to a new university.  This it appears that they have moved on from these two papers.   This is distressing:  they have not seen it through to actual publication.  (I understand now that arXiv does not count; being just a depository for papers that may not be peer-reviewed.)  Am I right in assuming that, being unpublished, it can't even be summarized in the Pythagorean Triple topic area?  This is doubly bad:  it is languishing on arXiv, but, as a precedent it prevents anyone else from claiming the topic.  
:I am presently fooling (sic!) with averaging Pythagorean triples; which is well off-topic, but interesting.  While doing that, I may have come up with a 'end-run' around O. & N. For the usual planar triangle, ABC.  Let a = q + r, b = r + s, 
:and c = q + r + s. If C is a right angle, then r^2 = 2sq.  Now back to the classic parameter 'n' and the uneven upstart 'u'.  The uneven u = 2t+1   (It is too confusing to use the 'm' that O. & N. use since it is not the 'm' with which we are very familiar.) 
:Replace q with u^2, r with 2un, and s with 2(n^2).   Q.E.D. (right?)    
:A chart may be generated for all integer values of 'n', and all odd integer values of 'u'.  This extends into all four quadrants, but only Quadrant I is used for Pythagorean triples.  I suggest that 'u' be the abscissa, and that 'n' be the ordinate.
:Mathematical feedback would be appreciated. ([[User talk:Lemchastain|talk]]) Ljc 20:47, 10 November 2022 (UTC)
::PS to above, also by Lem Chastain, as "Ljc"
::It occurred to me that some casual reader might suspect circular reasoning above, in assuming the expressions for a, b, and c.  Firstly, if there was a contradiction in the use of q, r, and s, then there would be no "counting number" solution for r^2 = 2sq.  Secondly, it was already known that there were numerous Pythagorean triples that satisfied those three expressions.  In fact, all of them satisfy the equations in terms of 'u' and 'n'.   [Note that the common 'm' and 'n' expressions can't:  m^2 - n^2, and m^2 + n^2 are only two terms, and 2mn only one term.]  ([[User talk:Lemchastain|talk]]) Ljc 15:40, 11 November 2022 (UTC)
:::1) Wikipedia does not cover unpublished papers and 2) Do not insert your personal comments into articles, as you did [https://en.wikipedia.org/w/index.php?title=Pythagorean_triple&diff=prev&oldid=1122804321 here]. Lemchastain, Wikipedia is simply not for the kinds of things you are trying to use it for. You should consider opening your own blog somewhere. [[User:MrOllie|MrOllie]] ([[User talk:MrOllie|talk]]) 22:01, 19 November 2022 (UTC)

== Update reference  ==

Please update the reference [26]

https://en.m.wikipedia.org/wiki/Pythagorean_triple#cite_note-Yekutieli-26

to the published version of the paper:

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Amnon Yekutieli

The American Mathematical Monthly 
Volume 130, 2023 - Issue 4

https://doi.org/10.1080/00029890.2023.2176114

 

https://en.m.wikipedia.org/wiki/Pythagorean_triple#cite_note-Yekutieli-26 [[User:Amyekut|Amyekut]] ([[User talk:Amyekut|talk]]) 13:50, 6 May 2023 (UTC)
:{{done}} —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 18:15, 6 May 2023 (UTC)