Talk:Pell's equation

Pell's equation has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it.
Review: September 3, 2020. (Reviewed version).

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A fact from Pell's equation appeared on Wikipedia's Main Page in the Did you know? column on 19 October 2020 (check views). The text of the entry was as follows:

Did you know... that the solution to Pell's equation was mistakenly attributed to mathematician John Pell?

A record of the entry may be seen at Wikipedia:Recent additions/2020/October. The nomination discussion and review may be seen at Template:Did you know nominations/Pell's equation.

Untitled[edit]

<quote> It turns out that if (p, q) satisfies Pell's equation, then so does (2pq, 2q^2-1).</quote>

Does it?

$n(2q^{2}-1)^{2}+1=4nq^{4}-4nq^{2}+n+1$ ; $4p^{2}q^{2}=4(nq^{2}+1)q^{2}=4nq^{4}+4q^{2}$

Which aren't generally equal. (Probably only for n = -1) --User:Xaos

Indeterminate equations[edit]

I removed the words "quadratic indeterminate" from

"Pell's equation is any quadratic indeterminate Diophantine equation of the form

x^{2}-ny^{2}=1

."

because the form already says that the equation is quadratic and indeterminate (in the sense of underdetermined). It is claimed that "there are many scholars referring to Pell's equation as an indeterminate equation", but I've seen no evidence for it and I still doubt that it's a standard term; anyway, it clearly is superfluous. -- Jitse Niesen (talk) 07:49, 10 April 2006 (UTC)[]

As motivation...[edit]

What is the purpose of this section with the square root of 2, and averaging the two fractions and so on. This method is not detailed like the Indian method or Lagrange's, and it only seems to work with this specific example. It provides no evidence to back the procedure up. Perhaps this section could be clarified, or maybe it should be considered for removal. —The preceding unsigned comment was added by Xcelerate (talk • contribs) 17:18, 15 January 2007 (UTC).[]

As part of a major reorganization of this article, I removed this section. Like you I found it unhelpful. I replaced it with a more straightforward worked-out example for n = 7, which gives I think a better flavor of the general technique. —David Eppstein 20:09, 20 March 2007 (UTC)[]

A (partial) solution given by Euler (??) was to write Pell's equation

$Nx^{2}-y^{2}=1$ as $({\frac {m}{n}})^{2}=(1/x^{2})+(y/x)^{2}$ then for big (x,y) y=m and x=n with m and n the convergents of the continued fraction for ${\sqrt {(}}N)$ —Preceding unsigned comment added by Karl-H (talk • contribs)

I believe the continued fraction solution technique described in the article is due to Euler. Is that what you mean? —David Eppstein 22:58, 24 March 2007 (UTC)[]

history[edit]

ok, i dont know about the indians, but it was certainly studied before Pell's times. Fermat certainly did. Fermat's theorem on Epll eqn.

16:48, 10 June 2007 (UTC)70.18.52.179 16:48, 10 June 2007 (UTC)[]

This really needs some more citations in the history section. Where's the source that Archimedes used it to get an approximation to the square root of 3? How do we know Greeks studied in the 5th century BCE? These are assumptions, possibly sound ones, but we need sources. Sceptic1954 (talk) 19:42, 13 May 2013 (UTC)[]

Claim about the Riemann hypothesis[edit]

"Gauss classified such solutions into 64 or 65 sets, with the precise classification of one or the other implying the truth or falsity of the Riemann hypothesis."

This seems too ridiculous to be true, even though it is from an old edit (Feb 2005) and one would think some expert would have noticed it by now. I don't believe Gauss could have possessed any mathematical statement that is equivalent to the Riemann hypothesis. (At best he could have guessed the precise error term for the Prime Number Theorem that is equivalent to RH, and he didn't.) It is not clear what the quoted sentence refers to as "such solutions", but anyway I don't think there exists any such statement known to be equivalent to RH. 128.36.156.146 (talk) 05:18, 1 December 2008 (UTC)[]

Comment[edit]

i have worked on equation of same type that is Nx^2 + k = y^2 , i have found that when N is not a perfect square, then y(2m) = {2({y(m)}^2} - k}/√k and x(2m) = 2x(m)× y(m); where m is the iteration number or the mth value for x and y , that is y(m) is the mth value for y and y(2m) is the 2×mth value for y. Ranjitr303 (talk) 06:48, 24 June 2010 (UTC)[]

Does this make sense to anyone?[edit]

The article is generally fine but I can't grasp the intent of

An alternative method to solving, once finding the first non-trivial solution, one could take the original equation $x^{2}-ny^{2}=1$ and factor the left hand side as a difference of squares, yielding $(x+y{\sqrt {n}})(x-y{\sqrt {n}})=1.$ Once in this form, one can simply raise each side of the equation to the kth power, and recombining the factored form to a single difference statement. The solution $s$ will be of the form $(x-s)^{k}+n*(y-s)^{k}=1.$

The fundamental solution is not used and the result is a big mess There are all kinds of true facts such as: If α is the fundamental solution then $\alpha ^{k}$ is very close to an even integer 2a (In fact it is very very close to 2a+1/2a or 2a-1/2a) and the kth solution is |a^2-nb^2|=1 where b is easy to find once one has a. (even with k=1, 8+3√7=15.937253933.. while 16-1/16=15.9375 and for bigger k it is even more dramatic ) BUT cute as that is, it is not a very effective computational method, so I don't advocate for it, just making the indicated two sentences clearer or killing them. Gentlemath (talk) 03:28, 16 July 2010 (UTC)[]

The passage makes no sense to me. It is unreferenced and was put in on 21 September 2009 by User:Cup of Calculus, who has never had any other content edits on Wikipedia except one other one that same day. I'll delete it. Loraof (talk) 15:57, 7 January 2016 (UTC)[]

The form a²x² + c = y² attributed to Diophantus[edit]

It is mentioned in the article that Diophantus solved an equation of the form a²x² + c = y² for a = 1 and c = -1, 1, 12, and for a = 3 and c = 9. Aren't all these cases quite trivial? Somehow it doesn't make sense to me that this is what he did, and even if so, that it is worth mentioning. 188.169.229.30 (talk) 10:33, 8 January 2012 (UTC)[]

Not when all numbers involved are integers (or, which is equivalent in non-triviality, rational). Which in this context they are.--Matt Westwood 21:36, 8 January 2012 (UTC)[]

Actually I think the integer case is trivial, at least with a = 1 — it's the rational case that isn't. My understanding is that Diophantus was concerned more with rational solutions than with integer solutions. —David Eppstein (talk) 22:32, 8 January 2012 (UTC)[]

Brahmagupta did not find a solution to Pell's formula[edit]

The whole part about Brahmagupta solving the formula 1000 years earlier is fishy - all of the Brahmagupta, Chakravala method and Bashkara II pages disagree with this. Brahmagupta found a partial solution, and didn't invent the charkravala method. — Preceding unsigned comment added by 194.126.175.154 (talk) 17:55, 4 December 2013 (UTC)[]

Misleading about timeline[edit]

From the paragraph 'Solutions/Quantum algorithms' in the article:

Hallgren (2007) showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time. Hallgren's algorithm, which can be interpreted as an algorithm for finding the group of units of a real quadratic number field, was extended to more general fields by Schmidt & Völlmer (2005).

It seems from this text that Schmidt & Völlmer used Hallgren work to extend it and produce a more generic solution but they published their algorithm before Hallgren ! Isn't it a bit strange ?

I double checked the abstracts and the paper of Schmidt & Völlmer does cite Hallgren works: Our algorithms generalize and improve upon Hallgren's work [9] for the one-dimensional case corresponding to real-quadratic fields.

And from Hallgren abstract it seems to really be this paper that is cited: The second problem we solve is the principal ideal problem in real quadratic number fields.

How is it possible ? Could the date not be the dates of the first publication ? — Preceding unsigned comment added by 2A01:E35:8A3A:9A80:1E4B:D6FF:FEBB:19BC (talk) 05:16, 25 April 2015 (UTC)[]

Assessment comment[edit]

The comment(s) below were originally left at Talk:Pell's equation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs a more encyclopedic style, I think. Also, is this yet a balanced account of the theory? Geometry guy 22:21, 9 June 2007 (UTC)[]

Last edited at 22:21, 9 June 2007 (UTC). Substituted at 02:27, 5 May 2016 (UTC)

Title[edit]

Notice: Better sooner or later, the title of the article must be changed to "Pell equation". The English version of wikipedia must avoid bad grammar in its titles.Highness 04:58, 1 July 2016 (UTC) — Preceding unsigned comment added by J20160628 (talk • contribs)

There is nothing ungrammatical about "Pell's equation". Your own grammar, however, is faulty: "Pell equation" cannot be used in that form without an article. —David Eppstein (talk) 17:44, 1 July 2016 (UTC)[]

GA Review[edit]

This review is transcluded from Talk:Pell's equation/GA1. The edit link for this section can be used to add comments to the review.

Reviewer: HeartGlow30797 (talk · contribs) 03:47, 21 August 2020 (UTC)[]

Rate	Attribute	Review Comment
1. Well written:
	1a. the prose is clear, concise, and understandable to an appropriately broad audience; spelling and grammar are correct.
	1b. it complies with the manual of style guidelines for lead sections, layout, words to watch, fiction, and list incorporation.
2. Verifiable with no original research:
	2a. it contains a list of all references (sources of information), presented in accordance with the layout style guideline.
	2b. all inline citations are from reliable sources, including those for direct quotations, statistics, published opinion, counter-intuitive or controversial statements that are challenged or likely to be challenged, and contentious material relating to living persons—science-based articles should follow the scientific citation guidelines.
	2c. it contains no original research.
	2d. it contains no copyright violations nor plagiarism.
3. Broad in its coverage:
	3a. it addresses the main aspects of the topic.	Excellent job on this.
	3b. it stays focused on the topic without going into unnecessary detail (see summary style).
	4. Neutral: it represents viewpoints fairly and without editorial bias, giving due weight to each.
	5. Stable: it does not change significantly from day to day because of an ongoing edit war or content dispute.	Note: Constant updates.
6. Illustrated, if possible, by media such as images, video, or audio:
	6a. media are tagged with their copyright statuses, and valid fair use rationales are provided for non-free content.	Note: The cover picture is an original work.
	6b. media are relevant to the topic, and have suitable captions.
	7. Overall assessment.	This is my first assessment and if you feel this is wrong, you should request a second opinion. Thank you!

Good Article review progress box

Criteria: 1a. prose () 1b. MoS () 2a. ref layout () 2b. cites WP:RS () 2c. no WP:OR () 2d. no WP:CV ()

3a. broadness () 3b. focus () 4. neutral () 5. stable () 6a. free or tagged images () 6b. pics relevant ()

Note: this represents where the article stands relative to the Good Article criteria. Criteria marked are unassessed

Comments: No MOS issues seen with fractions per MOS:FRAC. Further cmts to be made. Eumat114 (Message) 03:31, 22 August 2020 (UTC)[]

Quick comment: "compose" triples should not be changed to produce triples, because the sentence is talking about combining two triples to get new ones. Compare the other usages at Composition#Mathematics. XOR'easter (talk) 18:15, 22 August 2020 (UTC)[]

@HeartGlow30797: Two points with regards to the comments in 1(a):

"Brahmagupta solved many Pell equations with this method; in particular he showed how to obtain solutions..." I'm not like the word "in particular". We could replace it with "...this method by showing how to obtain..." — In fact I think this sentence is better as is;
"(sequence A001081 (x) and A001080 (y) in OEIS)." Provide a citation instead of putting it in parentheses. Do the same to all others. — The article is using the OEIS link template: Template:OEIS link, which is best practice.

And one wrt. point 3(b): rather than recommending shortnening, which is a difficult editorial instruction, I'm inclined to think you should identify a passage that you think would better be in another article and suggest a better home for the material. The point of WP:SUMMARY is not to have deprive readers of technical content, but rather to ensure that the material we have is digestable. (This is a shallow response to the review. I'm going to look over the article more carefully.) — Charles Stewart (talk) 11:24, 27 August 2020 (UTC)[]

Passed :D HeartGlow ^(talk) 10:05, 3 September 2020 (UTC)[]

Did you know nomination[edit]

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by SL93 (talk) 22:21, 10 October 2020 (UTC)
[]

( Review or comment
Article history )

... that the Pell's equation was mistakenly attributed to mathematician John Pell? (Source: ResearchGate paper and many others)

Improved to Good Article status by Eumat114 (talk). Self-nominated at 10:45, 4 September 2020 (UTC).[]

Comment (not a full review): Although this has been approved for GA status, it appears that it is not yet in compliance with DYK rules for sourcing, which require at least one reliable source in every paragraph (other than paragraphs in the lead of the article or in the lead of a section that summarize later sourced content). In particular, unsourced paragraphs include: the penultimate paragraph of "History" (the one with the hook claim! the note from the lead can be repeated here but it doesn't really contain a reliable source for the claim that Euler was mistaken, and the link you give in the nomination is not reliable), the first paragraph of "Fundamental solution via continued fractions", the first paragraph of "Concise representation and faster algorithms", the entire section "Continued fractions", the first paragraph of "The negative Pell equation" (note 3 is not a source). Incidentally (although this is not a DYK issue) I am a little surprised that the highly inconsistent reference formatting (where half the references are footnotes and the other half are inline parenthetical references, not counting the ones where the inline reference is also part of the text of the article) was allowed to pass GA. —David Eppstein (talk) 21:03, 4 September 2020 (UTC)[]

I am working on the sourcing, and as for the style I think using the footnote style is preferred. Eumat114 (Message) 07:55, 13 September 2020 (UTC)[]

I note what David Eppstein says which puts us DYK folks in a dilemma because we don't have the mathematical knowledge to properly assess the position. My view is that as a newly promoted GA, this nomination should be allowed to proceed, even if it is on an IAR basis. So I will review it. The article is new enough and long enough. The hook fact in the lead is cited inline, the article is neutral and I detected no copyright issues. No QPQ is needed here. Cwmhiraeth (talk) 06:08, 21 September 2020 (UTC)[]

It does not take any mathematical expertise or careful reading to look at the article and observe that it still has entire paragraphs without sources (at the ends of their sections so not summaries of anything later). —David Eppstein (talk) 06:46, 21 September 2020 (UTC)[]

I have to note that I will have more time to work on this in 1 or 2 days, and given the waiting time for DYK it probably won't take too long for it to work out.

Eumat114 (Message) 03:17, 22 September 2020 (UTC)[]

Return to WP:DYKN for further work. Yoninah (talk) 19:29, 22 September 2020 (UTC)[]

@All reviewers: I have fixed the essential sourcing (as much as I am able to), and is working on converting the refs to footnote. Could you please check? Eumat114 (Message) 00:47, 29 September 2020 (UTC)[]

Per earlier reviews and fixes to the article. --evrik ^(talk) 19:44, 10 October 2020 (UTC)[]

What is meant by "smallest solution"?[edit]

The section The smallest solution of Pell equations begins as follows:

"The following is a list of the smallest solution (fundamental solution) to $x^{2}-ny^{2}=1$ with n ≤ 128."

But the meaning of "smallest" solution is not explained.

Of course, if two solutions are (x,y) and (x',y') with x < x' and y < y' then it is natural to consider (x,y) to be a "smaller" solution than (x',y').

But what if, for instance, x < x' < y' < y ? (Or can this never happen?) 2601:200:C000:1A0:1860:7903:F5E8:6EE (talk) 21:31, 22 June 2021 (UTC)[]

it's the solution

x,y

for which the value of

x+y{\sqrt {n}}

is smallest. The more common term is "fundamental solution", which is defined in the article; I changed the section title accordingly. Thank you.--Qcomp (talk) 21:55, 22 June 2021 (UTC)[]

It is never going to be the case that a smaller x corresponds to a bigger y. Saying that

x+y{\sqrt {n}}

is smallest is making it unnecessarily complicated and WP:TECHNICAL. —David Eppstein (talk) 22:06, 22 June 2021 (UTC)[]

true, but I think the question was legitimate, as "smallest" for a pair (x,y) had not been defined. I gave the customary definition. But the article already defines the "fundamental solution" before the section that caused the confusion. Therefore I changed the section title and removed he notion of "smallest" (and also the sentence with the square root term, that I had added), as it is not needed. Hope that is both clearer and less technical. --Qcomp (talk) 22:26, 22 June 2021 (UTC)[]

Anh Duc Wiki -3

Sunday, October 3, 2021