Editing Pell's equation (section)

==History==
As early as 400 BC in [[India]] and [[Greece]], mathematicians studied the numbers arising from the ''n''&nbsp;=&nbsp;2 case of Pell's equation,
<math display="block">x^2 - 2 y^2 = 1,</math>
and from the closely related equation
<math display="block">x^2 - 2 y^2 = -1</math>
because of the connection of these equations to the [[square root of 2]].<ref name="knorr76"/> Indeed, if ''x'' and ''y'' are [[Natural number|positive integers]] satisfying this equation, then ''x''/''y'' is an approximation of {{math|{{sqrt|2}}}}. The numbers ''x'' and ''y'' appearing in these approximations, called [[Pell number|side and diameter numbers]], were known to the [[Pythagoreanism|Pythagoreans]], and [[Proclus]] observed that in the opposite direction these numbers obeyed one of these two equations.<ref name="knorr76"/> Similarly, [[Baudhayana]] discovered that ''x'' = 17, ''y'' = 12 and ''x'' = 577, ''y'' = 408 are two solutions to the Pell equation, and that 17/12 and 577/408 are very close approximations to the square root of 2.<ref>{{MacTutor Biography|id=Baudhayana|title=Baudhayana}}</ref>

Later, [[Archimedes]] approximated the [[square root of 3]] by the rational number 1351/780. Although he did not explain his methods, this approximation may be obtained in the same way, as a solution to Pell's equation.<ref name="knorr76">{{citation
 | last = Knorr | first = Wilbur R. | author-link = Wilbur Knorr
 | issue = 2
 | journal = Archive for History of Exact Sciences
 | mr = 0497462
 | pages = 115–140
 | title = Archimedes and the measurement of the circle: a new interpretation
 | volume = 15
 | year = 1976
 | doi=10.1007/bf00348496
 | s2cid = 120954547 }}.</ref>
Likewise, [[Archimedes's cattle problem]] &mdash; an ancient [[Word problem (mathematics education)|word problem]] about finding the number of cattle belonging to the sun god [[Helios]] &mdash; can be solved by reformulating it as a Pell's equation. The manuscript containing the problem states that it was devised by Archimedes and recorded in a letter to [[Eratosthenes]],<ref>{{cite journal |doi= 10.2307/2589706 |last= Vardi |first= I. |title= Archimedes' Cattle Problem |journal=[[American Mathematical Monthly]] |year= 1998 |volume= 105 |pages=305–319 | issue= 4 |publisher=  Mathematical Association of America |jstor= 2589706 |citeseerx= 10.1.1.33.4288 }}</ref> and the attribution to Archimedes is generally accepted today.<ref>{{cite book |title=Ptolemaic Alexandria |first=Peter M. |last=Fraser |author-link=Peter Fraser (classicist) |publisher=Oxford University Press |year=1972}}</ref><ref>{{cite book |title=Number Theory, an Approach Through History |first=André |last=Weil |author-link=André Weil |publisher=Birkhäuser |year=1972}}</ref>

Around AD 250, [[Diophantus]] considered the equation
<math display="block">a^2 x^2 + c = y^2,</math>
where ''a'' and ''c'' are fixed numbers, and ''x'' and ''y'' are the variables to be solved for.
This equation is different in form from Pell's equation but equivalent to it.
Diophantus solved the equation for (''a'',&nbsp;''c'') equal to (1,&nbsp;1), (1,&nbsp;−1), (1,&nbsp;12), and (3,&nbsp;9). [[Al-Karaji]], a 10th-century Persian mathematician, worked on similar problems to Diophantus.<ref>{{Cite journal |last=Izadi |first=Farzali |date=2015 |title=Congruent numbers via the Pell equation and its analogous counterpart |url=http://www.nntdm.net/papers/nntdm-21/NNTDM-21-1-70-78.pdf |journal=Notes on Number Theory and Discrete Mathematics |volume=21 |pages=70–78}}</ref>

In Indian mathematics, [[Brahmagupta]] discovered that
<math display="block">(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2,</math>
a form of what is now known as [[Brahmagupta's identity]]. Using this, he was able to "compose" triples <math>(x_1, y_1, k_1)</math> and <math>(x_2, y_2, k_2)</math> that were solutions of <math>x^2 - Ny^2 = k</math>, to generate the new triples
: <math>(x_1x_2 + Ny_1y_2 , x_1y_2 + x_2y_1 , k_1k_2)</math> and <math>(x_1x_2 - Ny_1y_2 , x_1y_2 - x_2y_1 , k_1k_2).</math>
Not only did this give a way to generate infinitely many solutions to <math>x^2 - Ny^2 = 1</math> starting with one solution, but also, by dividing such a composition by <math>k_1k_2</math>, integer or "nearly integer" solutions could often be obtained. For instance, for <math>N = 92</math>, Brahmagupta composed the triple (10,&nbsp;1,&nbsp;8) (since <math>10^2 - 92(1^2) = 8</math>) with itself to get the new triple (192,&nbsp;20,&nbsp;64). Dividing throughout by 64 ("8" for <math>x</math> and <math>y</math>) gave the triple (24,&nbsp;5/2,&nbsp;1), which when composed with itself gave the desired integer solution (1151,&nbsp;120,&nbsp;1). Brahmagupta solved many Pell's equations with this method, proving that it gives solutions starting from an integer solution of <math>x^2 - Ny^2 = k</math> for ''k'' = ±1, ±2, or ±4.<ref name=stillwell>{{citation | year=2002 | title = Mathematics and its history | author-link1=John Stillwell | author1=John Stillwell | edition=2nd | publisher=Springer | isbn=978-0-387-95336-6 | pages=72–76 | url=https://books.google.com/books?id=WNjRrqTm62QC&pg=PA72}}.</ref>

The first general method for solving the Pell's equation (for all ''N'') was given by [[Bhāskara&nbsp;II]] in 1150, extending the methods of Brahmagupta. Called the [[chakravala method|chakravala (cyclic) method]], it starts by choosing two relatively prime integers <math>a</math> and <math>b</math>, then composing the triple <math>(a, b, k)</math> (that is, one which satisfies <math>a^2 - Nb^2 = k</math>) with the trivial triple <math>(m, 1, m^2 - N)</math> to get the triple <math>\big(am + Nb, a + bm, k(m^2 - N)\big)</math>, which can be scaled down to
<math display="block">\left(\frac{am + Nb}{k}, \frac{a + bm}{k}, \frac{m^2 - N}{k}\right).</math>

When <math>m</math> is chosen so that <math>\frac{a + bm}{k}</math> is an integer, so are the other two numbers in the triple. Among such <math>m</math>, the method chooses one that minimizes <math>\frac{m^2 - N}{k}</math> and repeats the process. This method always terminates with a solution. Bhaskara used it to give the solution ''x''&nbsp;=&nbsp;{{val|1766319049}}, ''y''&nbsp;=&nbsp;{{val|226153980}} to the ''N''&nbsp;=&nbsp;61 case.<ref name=stillwell/>

Several European mathematicians rediscovered how to solve Pell's equation in the 17th century. [[Pierre de Fermat]] found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians.<ref>In February 1657, Pierre de Fermat wrote two letters about Pell's equation.  One letter (in French) was addressed to Bernard Frénicle de Bessy, and the other (in Latin) was addressed to Kenelm Digby, whom it reached via Thomas White and then William Brouncker.
* {{cite book |last1=Fermat |first1=Pierre de |editor1-last=Tannery |editor1-first=Paul |editor2-last=Henry |editor2-first=Charles |title=Oeuvres de Fermat |date=1894 |publisher=Gauthier-Villars et fils |location=Paris, France |pages=333–335 |volume=2nd vol. |url=https://www.biodiversitylibrary.org/item/62833#page/351/mode/1up |language=fr, la}} The letter to Frénicle appears on pp.&nbsp;333–334; the letter to Digby, on pp.&nbsp;334–335.
The letter in Latin to Digby is translated into French in:
* {{cite book |last1=Fermat |first1=Pierre de |editor1-last=Tannery |editor1-first=Paul |editor2-last=Henry |editor2-first=Charles |title=Oeuvres de Fermat |date=1896 |publisher=Gauthier-Villars et fils |location=Paris, France |pages=312–313 |volume=3rd vol. |url=https://www.biodiversitylibrary.org/item/62740#page/334/mode/1up |language=fr, la}}
Both letters are translated (in part) into English in:
* {{cite book |editor1-last=Struik |editor1-first=Dirk Jan |title=A Source Book in Mathematics, 1200–1800 |date=1986 |publisher=Princeton University Press |location=Princeton, New Jersey, USA |pages=29–30 |isbn=9781400858002 |url=https://books.google.com/books?id=o-3_AwAAQBAJ&pg=PA29}}</ref> In a letter to [[Kenelm Digby]], [[Bernard Frénicle de Bessy]] said that Fermat found the smallest solution for ''N'' up to 150 and challenged [[John Wallis]] to solve the cases ''N'' = 151 or 313. Both Wallis and [[William Brouncker, 2nd Viscount Brouncker|William Brouncker]] gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.<ref>In January 1658, at the end of {{lang|la|Epistola&nbsp;XIX}} (letter&nbsp;19), Wallis effusively congratulated Brouncker for his victory in a battle of wits against Fermat regarding the solution of Pell's equation. [https://books.google.com/books?id=_6nstlHZzaEC&pg=PA807 From p.&nbsp;807 of (Wallis, 1693)]: ''"Et quidem cum Vir Nobilissimus, utut hac sibi suisque tam peculiaria putaverit, & altis impervia, (''quippe non omnis fert omnia tellus'') ut ab Anglis haud speraverit solutionem; profiteatur tamen ''qu'il sera pourtant ravi d'estre destrompé par cet ingenieux & scavant Signieur''; erit cur & ipse tibi gratuletur.  Me quod attinet, humillimas est quod rependam gratias, quod in Victoriae tuae partem advocare dignatus es,&nbsp;..."'' (And indeed, Most Noble Sir [i.e., Viscount Brouncker], he [i.e., Fermat] might have thought [to have] all to himself such an esoteric [subject, i.e., Pell's equation] with its impenetrable profundities (''for all land does not bear all things'' [i.e., not every nation can excel in everything]), so that he might hardly have expected a solution from the English; nevertheless, he avows ''that he will, however, be thrilled to be disabused by this ingenious and learned Lord'' [i.e., Brouncker]; it will be for that reason that he [i.e., Fermat] himself would congratulate you.  Regarding myself, I requite with humble thanks your having deigned to call upon me to take part in your Victory,&nbsp;...) Note: The date at the end of Wallis' letter is "Jan.&nbsp;20, 1657"; however, that date was according to the old Julian calendar that Britain finally [[Calendar (New Style) Act 1750|discarded in 1752]]: most of the rest of Europe would have regarded that date as January&nbsp;31, 1658. See [[Old Style and New Style dates#Transposition of historical event dates and possible date conflicts]].</ref>

[[John Pell (mathematician)|John Pell]]'s connection with the equation is that he revised [[Thomas Branker]]'s translation<ref>{{citation |last=Rahn |first=Johann Heinrich |title=An introduction to algebra |url=http://eebo.chadwyck.com/search/full_rec?SOURCE=pgimages.cfg&ACTION=ByID&ID=V57365 |year=1668 |editor-last=Brancker |editor-first=Thomas |orig-year=1659 |editor2-last=Pell}}.</ref> of [[Johann Rahn]]'s 1659 book ''Teutsche Algebra''<ref group=note>''[[wikt:Special:Search/Teutsch|Teutsch]]'' is an obsolete form of ''Deutsch'', meaning "German". Free E-book: [https://books.google.com/books?id=ZJg_AAAAcAAJ ''Teutsche Algebra''] at Google Books.</ref> into English, with a discussion of Brouncker's solution of the equation. [[Leonhard Euler]] mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell.<ref name=":1">{{Cite book |last=Tattersall |first=James |url=https://pdfs.semanticscholar.org/6881/a5169f76c5b4de2b206346815313f343af52.pdf |archive-url=https://web.archive.org/web/20200215183051/https://pdfs.semanticscholar.org/6881/a5169f76c5b4de2b206346815313f343af52.pdf |url-status=dead |archive-date=2020-02-15 |title=Elementary Number Theory in Nine Chapters |publisher=Cambridge |year=2000 |pages=274 |doi=10.1017/CBO9780511756344 |isbn=9780521850148 |s2cid=118948378}}</ref>

The general theory of Pell's equation, based on [[continued fraction]]s and algebraic manipulations with numbers of the form <math>P + Q\sqrt{a},</math> was developed by Lagrange in 1766–1769.<ref>"Solution d'un Problème d'Arithmétique", in [[Joseph Alfred Serret]] (ed.), [http://gdz.sub.uni-goettingen.de/dms/load/toc/?PID=PPN308899644 ''Œuvres de Lagrange'', vol.&nbsp;1], pp.&nbsp;671–731, 1867.</ref> In particular, Lagrange gave a proof that the Brouncker–Wallis algorithm always terminates.