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{{Use dmy dates|date=October 2019}}
[[File:Pell's equation.svg|thumb|360px|Pell's equation for ''n''&nbsp;=&nbsp;2 and six of its integer solutions]]
'''Pell's equation''' (also called the '''Pell–Fermat equation''') is any [[Diophantine equation]] of the form

:<math>x^2-ny^2=1\,</math>

where ''n'' is a given positive [[Square number|nonsquare]] [[integer]] and integer solutions are sought for ''x'' and ''y''. In [[Cartesian coordinates]], the equation has the form of a [[hyperbola]]; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the [[Triviality (mathematics)|trivial solution]] with ''x''&nbsp;=&nbsp;1 and ''y''&nbsp;=&nbsp;0.  [[Joseph Louis Lagrange]] proved that, as long as ''n'' is not a [[square number|perfect square]], Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately [[Diophantine approximation|approximate]] the [[square root]] of&nbsp;''n'' by [[rational number]]s of the form&nbsp;''x/y''.

This equation was first studied extensively [[Indian mathematics|in India]], starting with [[Brahmagupta]], who developed the [[chakravala method|''chakravala'' method]] to solve Pell's equation and other quadratic indeterminate equations in his [[Brahmasphutasiddhanta|''Brahma Sphuta Siddhanta'']] in 628, about a thousand years before Pell's time. His ''Brahma Sphuta Siddhanta'' was translated into [[Arabic]] in 773 and was subsequently translated into [[Latin]] in 1126. [[Bhaskara II]] in the 12th century and [[Narayana Pandit]] in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Solutions to specific examples of Pell's equation, such as the [[Pell number]]s arising from the equation with ''n'' = 2, had been known for much longer, since the time of [[Pythagoras]] in [[Greek mathematics|Greece]] and to a similar date in India. The name of Pell's equation arose from [[Leonhard Euler]] mistakenly attributing [[Lord Brouncker]]'s solution of the equation to [[John Pell]].<ref>As early as 1732–1733 Euler believed that John Pell had developed a method to solve Pell's equation, even though Euler knew that Wallis had developed a method to solve it (although William Brouncker had actually done most of the work):
* {{cite journal |last1=Euler |first1=Leonard |title=De solutione problematum Diophantaeorum per numeros integros |journal=Commentarii Academiae Scientiarum Imperialis Petropolitanae (Memoirs of the Imperial Academy of Sciences at St. Petersburg) |date=1732–1733 |volume=6 |pages=175–188 |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015065400890;view=1up;seq=185 |trans-title=On the solution of Diophantine problems by integers}}  From p. 182: ''"At si ''a'' huiusmodi fuerit numerus, qui nullo modo ad illas formulas potest reduci, peculiaris ad invenienda ''p'' et ''q'' adhibenda est methodus, qua olim iam usi sunt ''Pellius'' et ''Fermatius''."'' (But if such an ''a'' be a number that can be reduced in no way to these formulas, the specific method for finding ''p'' and ''q'' is applied which ''Pell'' and ''Fermat'' have used for some time now.) From p. 183: ''"§. 19. Methodus haec extat descripta in operibus ''Wallisii'', et hanc ob rem eam hic fusius non-expono."'' (§. 19. This method exists described in the works of Wallis, and for this reason I do not present it here in more detail.)
* Lettre IX.  Euler à Goldbach, dated 10 August 1750 in:  {{cite book |editor1-last=Fuss |editor1-first=P.H. |title=Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … |trans-title=Mathematical and physical correspondence of some famous geometers of the 18th century … |date=1843 |location=St. Petersburg, Russia |page=37 |url=https://books.google.com/books?id=gf1OEXIQQgsC&pg=PA37#v=onepage&q&f=false |language=French, Latin, German}}  From page 37:  ''"Pro hujusmodi quaestionibus solvendis excogitavit D. Pell Anglus peculiarem methodum in Wallisii operibus expositam."'' (For solving such questions, the Englishman Dr. Pell devised a singular method [which is] shown in Wallis' works.)
* {{cite book |last1=Euler |first1=Leonard |title=Vollständige Anleitung zur Algebra, II. Theil |trans-title=Complete Introduction to Algebra, Part 2 |date=1771 |publisher=St. Petersburg, Russia |location=Kayserlichen Akademie der Wissenschaften (Imperial Academy of Sciences) |page=227 |url=https://archive.org/stream/bub_gb_nfYGAAAAcAAJ#page/n503 |language=German}} From p. 227: ''"§98. Hierzu hat vormals ein gelehrter Engländer, Namens Pell, eine ganz sinnreiche Methode erfunden, welche wir hier erklären wollen."'' (§.98 Concerning this, a learned Englishman by the name of Pell has previously found a quite ingenious method, which we will explain here.)
* English translation: {{cite book |last1=Euler |first1=Leonard |title=Elements of Algebra … |date=1810 |publisher=J. Johnson |location=London, England |page=78 |edition=2nd |volume= 2nd vol. |url=https://books.google.com/books?id=mqI-AAAAYAAJ&pg=PA78#v=onepage&q&f=false}}
* {{cite book |last1=Heath |first1=Thomas L. |title=Diophantus of Alexandria : A Study in the History of Greek Algebra |date=1910 |publisher=Cambridge University Press |location=Cambridge, England |page=286 |url=https://archive.org/stream/diophantusofalex00heatiala#page/286 }} See especially footnote 4.</ref><ref group=note>In Euler's ''Vollständige Anleitung zur Algebra'' (pp. 227 ff), he presents a solution to Pell's equation which was taken from John Wallis' ''Commercium epistolicum'', specifically, Letter 17 (''Epistola XVII'') and Letter 19 (''Epistola XIX'') of:
* {{cite book |editor1-last=Wallis |editor1-first=John |title=Commercium epistolicum, de Quaestionibus quibusdam Mathematicis nuper habitum. |trans-title=Correspondence, about some mathematical inquiries recently undertaken |date=1658 |publisher=A. Lichfield |location=Oxford, England |url=https://books.google.com/books?id=RqFGAAAAcAAJ&pg=PA56#v=onepage&q&f=false |language=English, Latin, French}} The letters are in Latin.  Letter 17 appears on pp. 56–72.  Letter 19 appears on pp. 81–91.
*  French translations of Wallis' letters:  {{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  |volume=3rd vol. |url=https://www.biodiversitylibrary.org/item/62740#page/479/mode/1up |language=French, Latin}} Letter 17 appears on pp. 457–480.  Letter 19 appears on pp. 490–503.
Wallis' letters showing a solution to the Pell's equation also appear in volume 2 of Wallis' ''Opera mathematica'' (1693), which includes articles by John Pell:
*  {{cite book |last1=Wallis |first1=John |title=Opera mathematica: de Algebra Tractatus; Historicus & Practicus |trans-title=Mathematical works: Treatise on Algebra; historical and as presently practiced |volume=2nd vol. |date=1693 |location=Oxford, England |url=https://books.google.com/books?id=_6nstlHZzaEC&pg=PP15#v=onepage&q&f=false |language=Latin, English, French}} Letter 17 is on pp. 789–798 ; letter 19 is on pp. 802–806.  See also Pell's articles, where Wallis mentions (pp. 235, 236, 244) that Pell's methods are applicable to the solution of Diophantine equations:
:*  ''De Algebra D. Johannis Pellii; & speciatim de Problematis imperfecte determinatis.''  (On Algebra by Dr. John Pell and especially on an incompletely determined problem), pp. 234–236.
:*  ''Methodi Pellianae Specimen.'' (Example of Pell's method), pp. 238–244.
:*  ''Specimen aliud Methodi Pellianae.'' (Another example of Pell's method), pp. 244–246.
See also:
*  Whitford, Edward Everett (1912) "The Pell equation," doctoral thesis, Columbia University (New York, New York, USA), [https://archive.org/stream/pellequation00whitgoog#page/n63 p. 52.]
*  {{cite book |last1=Heath |first1=Thomas L. |title=Diophantus of Alexandria : A Study in the History of Greek Algebra |date=1910 |publisher=Cambridge University Press |location=Cambridge, England |page=286 |url=https://archive.org/stream/diophantusofalex00heatiala#page/286 }}</ref>

==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> x^2 - 2y^2=1 </math>
and from the closely related equation
:<math> x^2 - 2y^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.

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. | authorlink = 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}}.</ref>
[[Archimedes's cattle problem]] involves solving a Pellian equation. It is now generally accepted that this problem is due to Archimedes.<ref>{{cite book|title=Ptolemaic Alexandria|first=Peter M.|last=Fraser|authorlink=Peter Fraser (classicist)|publisher=Oxford University Press|year=1972}}</ref><ref>{{cite book|title=Number Theory, an Approach Through History|first=André|last=Weil|authorlink=André Weil|publisher=Birkhäuser|year=1972}}</ref>

Around AD 250, [[Diophantus]] considered the equation

:<math> 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'', ''c'') equal to (1, 1), (1, −1), (1, 12), and (3, 9). [[Al-Karaji]], a 10th-century Persian mathematician, worked on similar problems to Diophantus.

In Indian mathematics, [[Brahmagupta]] discovered that
:<math>(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>
(see [[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, 1, 8) (since <math>10^2 - 92(1^2) = 8</math>) with itself to get the new triple (192, 20, 64). Dividing throughout by 64 ('8' for <math>x</math> and <math>y</math>) gave the triple (24, 5/2, 1), which when composed with itself gave the desired integer solution (1151, 120, 1). Brahmagupta solved many Pell equations with this method; in particular he showed how to obtain 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 | authorlink1=John Stillwell | author1=John Stillwell | edition=2nd | publisher=Springer | isbn=978-0-387-95336-6 | pages=72–76 | url=https://books.google.com/?id=WNjRrqTm62QC&pg=PA72}}</ref>

The first general method for solving the Pell equation (for all ''N'') was given by [[Bhāskara 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>(am + Nb, a+bm, k(m^2-N))</math>, which can be scaled down to

:<math>\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 (proved by [[Joseph-Louis Lagrange]] in 1768). Bhaskara used it to give the solution ''x'' = 1766319049, ''y'' =&nbsp;226153980 to the notorious ''N'' = 61 case.<ref name=stillwell/>

Several European mathematicians rediscovered how to solve Pell's equation in the 17th century, apparently unaware that it had been solved almost five hundred years earlier in India. [[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=French, Latin}} The letter to Frénicle appears on pp. 333–334; the letter to Digby, on pp. 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=French, Latin}}
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 |url=https://books.google.com/books?id=o-3_AwAAQBAJ&pg=PA29#v=onepage&q&f=false}}</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 ''Epistola XIX'' (letter 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#v=onepage&q&f=false From p. 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, … "'' (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, … )  [Note: The date at the end of Wallis' letter is "Jan. 20. 1657"; however, that date was according to the old Julian calendar; when Britain finally [[Calendar (New Style) Act 1750|adopted the Gregorian calendar in 1751]], that date became January 30, 1658.]</ref>

[[John Pell]]'s connection with the equation is that he revised [[Thomas Branker]]'s  translation {{harv|Rahn|1668}} of [[Johann Rahn]]'s 1659 book "Teutsche Algebra"<ref group=note>"Teutsch" is a now obsolete referral to the German language, and here also to the, considered genuine, German culture area. Free E-book: [https://books.google.com/books?id=ZJg_AAAAcAAJ ''Teutsche Algebra'' (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.

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. 1, pp. 671–731, 1867.]</ref>

==Solutions==

===Fundamental solution via continued fractions===

Let <math>\tfrac{h_i}{k_i}</math> denote the sequence of [[Convergent (continued fraction)|convergents]] to the [[continued fraction|regular continued fraction]] for <math>\sqrt{n}</math>. This sequence is unique. Then the pair (''x''<sub>1</sub>,''y''<sub>1</sub>) solving Pell's equation and minimizing ''x'' satisfies ''x''<sub>1</sub> = ''h<sub>i</sub>'' and ''y''<sub>1</sub> = ''k<sub>i</sub>'' for some ''i''. This pair is called the ''fundamental solution''. Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

As {{harvtxt|Lenstra|2002}} describes, the time for finding the fundamental solution using the continued fraction method, with the aid of the [[Schönhage–Strassen algorithm]] for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair (''x''<sub>1</sub>,''y''<sub>1</sub>). However, this is not a [[polynomial time algorithm]] because the number of digits in the solution may be as large as {{radic|''n''}}, far larger than a polynomial in the number of digits in the input value ''n'' {{harv|Lenstra|2002}}.

===Additional solutions from the fundamental solution===
Once the fundamental solution is found, all remaining solutions may be calculated algebraically from
:<math> x_k + y_k \sqrt n = (x_1 + y_1 \sqrt n)^k,</math>
expanding the right side, [[equating coefficients]] of <math>\sqrt{n}</math> on both sides, and equating the other terms on both sides. This yields the [[recurrence relation]]s
:<math>\displaystyle x_{k+1} = x_1 x_k + n y_1 y_k,</math>
:<math>\displaystyle y_{k+1} = x_1 y_k + y_1 x_k.</math>

===Concise representation and faster algorithms===
Although writing out the fundamental solution (''x''<sub>1</sub>, ''y''<sub>1</sub>) as a pair of binary numbers may require a large number of bits, it may in many cases be represented more compactly in the form
:<math>x_1+y_1\sqrt n = \prod_{i=1}^t (a_i + b_i\sqrt n)^{c_i}</math>
using much smaller integers ''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>, and ''c''<sub>''i''</sub>.

For instance, [[Archimedes' cattle problem]] is equivalent to the Pell equation <math>x^2-410286423278424y^2=1</math>, the fundamental solution of which has 206545 digits if written out explicitly. However, the solution is also equal to
:<math>x_1+y_1\sqrt n=u^{2329},</math>
where
:<math>u = x'_1+y'_1\sqrt{4729494} = (300426607914281713365\sqrt{609}+84129507677858393258\sqrt{7766})^2</math> 
and <math>x'_1</math> and <math>y'_1</math> only have 45 and 41 decimal digits, respectively.<ref>{{harv|Lenstra|2002}}</ref>

Methods related to the [[quadratic sieve]] approach for [[integer factorization]] may be used to collect relations between prime numbers in the number field generated by {{radic|''n''}}, and to combine these relations to find a product representation of this type. The resulting algorithm for solving Pell's equation is more efficient than the continued fraction method, though it still takes more than polynomial time. Under the assumption of the [[generalized Riemann hypothesis]], it can be shown to take time
:<math>\exp O(\sqrt{\log N\log\log N}),</math>
where ''N''&nbsp;=&nbsp;log&nbsp;''n'' is the input size, similarly to the quadratic sieve {{harv|Lenstra|2002}}.

===Quantum algorithms===
{{harvtxt|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 {{harvtxt|Schmidt|Völlmer|2005}}.

==Example==
As an example, consider the instance of Pell's equation for ''n'' = 7; that is,
:<math>x^2-7y^2 = 1</math>
The sequence of convergents for the square root of seven are
:{| class="wikitable" style="text-align:center;"
|-
! ''h''&nbsp;/&nbsp;''k'' (Convergent)
! ''h''<sup>2</sup>&nbsp;−7''k''<sup>2</sup> (Pell-type approximation)
|-
| 2&nbsp;/&nbsp;1
| −3
|-
| 3&nbsp;/&nbsp;1
| +2
|-
| 5&nbsp;/&nbsp;2
| −3
|-
| 8&nbsp;/&nbsp;3
| +1
|}

Therefore, the fundamental solution is formed by the pair (8, 3). Applying the recurrence formula to this solution generates the infinite sequence of solutions

:(1, 0); (8, 3); (127, 48); (2024, 765); (32257, 12192); (514088, 194307); (8193151, 3096720); (130576328, 49353213); ... (sequence {{OEIS link|id=A001081}} (''x'') and {{OEIS link|id=A001080}} (''y'') in [[OEIS]])

The smallest solution can be very large. For example, the smallest solution to <math>x^2-313y^2 = 1</math> is (32188120829134849, 1819380158564160), and this is the equation which Frenicle challenged Wallis to solve.<ref>[http://primes.utm.edu/curios/page.php?short=313 Prime Curios!: 313]</ref> Values of ''n'' such that the smallest solution of <math>x^2-ny^2 = 1</math> is greater than the smallest solution for any smaller value of ''n'' are
:1, 2, 5, 10, 13, 29, 46, 53, 61, 109, 181, 277, 397, 409, 421, 541, 661, 1021, 1069, 1381, 1549, 1621, 2389, 3061, 3469, 4621, 4789, 4909, 5581, 6301, 6829, 8269, 8941, 9949, ... {{OEIS|id=A033316}}
(For these records, see {{oeis|id=A033315}} (''x''), and {{oeis|id=A033319}} (''y'')).

==The smallest solution of Pell equations==

The following is a list of the smallest solution to <math>x^2 - ny^2 = 1</math> with ''n'' ≤ 128. For square ''n'', there is no solution except (1, 0). (sequence {{OEIS link|id=A002350}} (''x'') and {{OEIS link|id=A002349}} (''y'') in [[OEIS]], or {{oeis|id=A033313}} (''x'') and {{oeis|id=A033317}} (''y'') (for nonsquare ''n''))

:{|class="wikitable"
!''n''
!''x''
!''y''
!''n''
!''x''
!''y''
!''n''
!''x''
!''y''
!''n''
!''x''
!''y''
|-
|1
|&#045;
|&#045;
|33
|23
|4
|65
|129
|16
|97
|62809633
|6377352
|-
|2
|3
|2
|34
|35
|6
|66
|65
|8
|98
|99
|10
|-
|3
|2
|1
|35
|6
|1
|67
|48842
|5967
|99
|10
|1
|-
|4
|&#045;
|&#045;
|36
|&#045;
|&#045;
|68
|33
|4
|100
|&#045;
|&#045;
|-
|5
|9
|4
|37
|73
|12
|69
|7775
|936
|101
|201
|20
|-
|6
|5
|2
|38
|37
|6
|70
|251
|30
|102
|101
|10
|-
|7
|8
|3
|39
|25
|4
|71
|3480
|413
|103
|227528
|22419
|-
|8
|3
|1
|40
|19
|3
|72
|17
|2
|104
|51
|5
|-
|9
|&#045;
|&#045;
|41
|2049
|320
|73
|2281249
|267000
|105
|41
|4
|-
|10
|19
|6
|42
|13
|2
|74
|3699
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==Connections==
Pell's equation has connections to several other important subjects in mathematics.

===Algebraic number theory===
Pell's equation is closely related to the theory of [[algebraic number]]s, as the formula
:<math>x^2 - n y^2 = (x + y\sqrt n)(x - y\sqrt n)</math>
is the [[Field norm|norm]] for the [[Ring (mathematics)|ring]] <math>\mathbb{Z}[\sqrt{n}]</math> and for the closely related [[quadratic field]] <math>\mathbb{Q}(\sqrt{n})</math>. Thus, a pair of integers <math>(x, y)</math> solves Pell's equation if and only if <math>x + y \sqrt{n}</math> is a [[Unit (ring theory)|unit]] with norm 1 in <math>\mathbb{Z}[\sqrt{n}]</math>. [[Dirichlet's unit theorem]], that all units of <math>\mathbb{Z}[\sqrt{n}]</math> can be expressed as powers of a single [[Fundamental unit (number theory)|fundamental unit]] (and multiplication by a sign), is an algebraic restatement of the fact that all solutions to the Pell equation can be generated from the fundamental solution. The fundamental unit can in general be found by solving a Pell-like equation but it does not always correspond directly to the fundamental solution of Pell's equation itself, because the fundamental unit may have norm −1 rather than 1 and its coefficients may be half integers rather than integers.

===Chebyshev polynomials===
Demeyer (2007) mentions a connection between Pell's equation and the [[Chebyshev polynomials]]:
If ''T<sub>i</sub>''&nbsp;(''x'') and ''U<sub>i</sub>''&nbsp;(''x'') are the Chebyshev polynomials of the first and second kind, respectively, then these polynomials satisfy a form of Pell's equation in any [[polynomial ring]] ''R''[''x''], with ''n'' = ''x''<sup>2</sup>&nbsp;−&nbsp;1:

:<math>T_i^2 - (x^2-1) U_{i-1}^2 = 1. \, </math>

Thus, these polynomials can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:

:<math>T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i. \, </math>

It may further be observed that, if (''x<sub>i</sub>'',''y<sub>i</sub>'') are the solutions to any integer Pell equation, then ''x<sub>i</sub>'' = ''T<sub>i</sub>'' (''x''<sub>1</sub>) and ''y<sub>i</sub>'' = ''y''<sub>1</sub>''U''<sub>''i''&nbsp;−&nbsp;1</sub>(''x''<sub>1</sub>) (Barbeau, chapter 3).

===Continued fractions===
A general development of solutions of Pell's equation <math>x^2-ny^2 = 1</math> in terms of [[continued fraction]]s of <math>\sqrt{n}</math> can be presented, as the solutions ''x'' and ''y'' are approximates to the square root of ''n'' and thus are a special case of continued fraction approximations for [[quadratic irrational]]s.

The relationship to the continued fractions implies that the solutions to Pell's equation form a [[semigroup]] subset of the [[modular group]]. Thus, for example, if ''p'' and ''q'' satisfy Pell's equation, then

:<math>\begin{pmatrix} p & q \\ nq & p \end{pmatrix}</math>

is a matrix of unit [[determinant]].  Products of such matrices take exactly the same form, and thus all such products yield solutions to Pell's equation. This can be understood in part to arise from the fact that successive convergents of a continued fraction share the same property: If ''p''<sub>''k''−1</sub>/''q''<sub>''k''−1</sub> and ''p''<sub>''k''</sub>/''q''<sub>''k''</sub> are two successive convergents of a continued fraction, then the matrix

:<math>\begin{pmatrix} p_{k-1} & p_{k} \\ q_{k-1} & q_{k} \end{pmatrix}</math>

has determinant (−1)<sup>''k''</sup>.

[[Størmer's theorem]] applies Pell equations to find pairs of consecutive [[smooth number]]s. As part of this theory, [[Carl Størmer|Størmer]] also investigated divisibility relations among solutions to Pell's equation; in particular, he showed that each solution other than the fundamental solution has a [[prime factor]] that does not divide&nbsp;''n''.

As Lenstra (2002) describes, Pell's equation can also be used to solve [[Archimedes' cattle problem]].

==The negative Pell equation==
The negative Pell equation is given by

:<math> x^2 - ny^2 = -1 </math>

It has also been extensively studied; it can be solved by the same method of continued fractions and will have solutions if and only if the period of the continued fraction has odd length. However it is not known which roots have odd period lengths and therefore not known when the negative Pell equation is solvable.  A necessary (but not sufficient) condition for solvability is that ''n'' is not divisible by 4 or by a prime of form 4''k''&nbsp;+&nbsp;3.<ref group=note>This is because the Pell equation implies that −1 is a [[quadratic residue]] modulo ''n''.</ref> Thus, for example, ''x''<sup>2</sup>&nbsp;−&nbsp;3''ny''<sup>2</sup>&nbsp;=&nbsp;−1 is never solvable, but ''x''<sup>2</sup>&nbsp;−&nbsp;5''ny''<sup>2</sup>&nbsp;=&nbsp;−1 may be.

The first few numbers ''n'' for which ''x''<sup>2</sup>&nbsp;−&nbsp;''ny''<sup>2</sup>&nbsp;=&nbsp;−1 is solvable are
:1, 2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, ... {{OEIS|id=A031396}}.

{{harvtxt|Cremona|Odoni|1989}} demonstrate that the proportion of square-free ''n'' divisible by ''k'' primes of the form 4''m''&nbsp;+&nbsp;1 for which the negative Pell equation is solvable is at least 40%. If the negative Pell equation does have a solution for a particular ''n'', its fundamental solution leads to the fundamental one for the positive case by squaring both sides of the defining equation:

:<math> (x^2 - ny^2)^2 = (-1)^2 </math>

implies

:<math> (x^2 + ny^2)^2 - n(2xy)^2 = 1.</math>

==Transformations==
The related equation

:<math> u^2 - dv^2 = \pm 2</math>  (*)

can be used to find solutions to the positive Pell equation for certain&nbsp;''d''.  [[Adrien-Marie Legendre|Legendre]] proved that all primes of form ''d''&nbsp;=&nbsp;4''m''&nbsp;+&nbsp;3 solve one case of (*), with the form 8''m''&nbsp;+&nbsp;3 solving the negative, and 8''m''&nbsp;+&nbsp;7 for the positive.  Their fundamental solution then leads to the one for ''x''<SUP>2</SUP>−''dy''<SUP>2</SUP>&nbsp;=&nbsp;1.  This can be shown by squaring both sides of (*)

:<math> (u^2 - dv^2)^2 = (\pm 2)^2 </math>

to get

:<math> (u^2 + dv^2)^2 - d(2uv)^2 = 4. </math>

Since <math> dv^2 = u^2 \mp 2 </math> from (*), it follows that

:<math> (u^2 \mp 1)^2 - d(uv)^2 = 1, </math>

and so the fundamental solutions to (*) are smaller than those of the associated negative Pell equation. For example, u<SUP>2</SUP>-3v<SUP>2</SUP> = -2 is {''u'',''v''} = {1, 1}, so ''x''<SUP>2</SUP>&nbsp;−&nbsp;3''y''<SUP>2</SUP>&nbsp;=&nbsp;1 has {''x'',''y''} = {2, 1}.  On the other hand, ''u''<SUP>2</SUP>&nbsp;−&nbsp;7''v''<SUP>2</SUP> =&nbsp;2 is {''u'',''v''} = {3,1}, so ''x''<SUP>2</SUP>&nbsp;−&nbsp;7y<SUP>2</SUP> = 1 has {''x'',''y''}&nbsp;=&nbsp;{8,3}.

Another related equation,

:<math> u^2 - dv^2 = \pm 4 </math>

can also be used to find solutions to Pell equations for certain ''d'', this time for the positive and negative case.  For the following transformations,<ref>[http://sites.google.com/site/tpiezas/008  A Collection of Algebraic Identities: Pell Equations.]</ref> if fundamental {''u'',''v''} are both [[odd number|odd]], then it leads to fundamental {x,y}.

1. If u<SUP>2</SUP>&nbsp;−&nbsp;dv<SUP>2</SUP> = −4, and {x,y} = {(''u''<SUP>2</SUP>&nbsp;+&nbsp;3)''u''/2, (''u''<SUP>2</SUP>&nbsp;+&nbsp;1)''v''/2}, then ''x''<SUP>2</SUP>&nbsp;−&nbsp;''dy''<SUP>2</SUP>&nbsp;=&nbsp;−1.

Ex. Let ''d'' = 13, then {''u'',''v''} = {3, 1} and {''x'',''y''} = {18, 5}.

2. If ''u''<SUP>2</SUP>&nbsp;−&nbsp;''dv''<SUP>2</SUP> =&nbsp;4, and {''x'',''y''} = {(''u''<SUP>2</SUP>&nbsp;−&nbsp;3)''u''/2, (''u''<SUP>2</SUP>&nbsp;−&nbsp;1)''v''/2}, then ''x''<SUP>2</SUP>&nbsp;−&nbsp;''dy''<SUP>2</SUP>&nbsp;=&nbsp;1.

Ex. Let ''d'' = 13, then {u,v} = {11, 3} and {x,y} = {649, 180}.

3. If ''u''<SUP>2</SUP>&nbsp;−&nbsp;''dv''<SUP>2</SUP> = −4, and {''x'',''y''} = {(''u''<SUP>4</SUP>&nbsp;+&nbsp;4''u''<SUP>2</SUP>&nbsp;+&nbsp;1)(''u''<SUP>2</SUP>&nbsp;+&nbsp;2)/2, (''u''<SUP>2</SUP>&nbsp;+&nbsp;3)(''u''<SUP>2</SUP>&nbsp;+&nbsp;1)''uv''/2}, then ''x''<SUP>2</SUP>&nbsp;−&nbsp;''dy''<SUP>2</SUP>&nbsp;=&nbsp;1.

Ex. Let ''d'' = 61, then {''u'',''v''} = {39, 5} and {''x'',''y''} = {1766319049, 226153980}.

Especially for the last transformation, it can be seen how solutions to {''u'',''v''} are ''much'' smaller than {''x'',''y''}, since the latter are [[sextic]] and [[quintic]] polynomials in terms of&nbsp;''u''.

==Notes==
{{reflist|group=note}}

==References==
<references/>

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==Further reading==
* {{cite book | zbl=1043.11027 | last=Williams | first=H. C. | chapter=Solving the Pell equation | pages=325–363 | editor1-last=Bennett | editor1-first=M. A. | editor2-last=Berndt | editor2-first=B.C. | editor2-link=Bruce C. Berndt | editor3-last=Boston | editor3-first=N. | editor3-link=Nigel Boston | editor4-last=Diamond | editor4-first=H.G. | editor5-last=Hildebrand | editor5-first=A.J. | editor6-last=Philipp | editor6-first=W. | title=Surveys in number theory: Papers from the millennial conference on number theory | location=Natick, MA | publisher=A K Peters | year=2002 | isbn=1-56881-162-4 }}

==External links==
* {{mathworld|urlname=PellEquation|title=Pell's equation}}
*[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html Pell's equation]
*[http://www.jakebakermaths.org.uk/maths/jshtmlpellsolverbigintegerv10.html Pell equation solver (''n'' has no upper limit)]
*[http://www.numbertheory.org/php/pell.html Pell equation solver (''n''<10^10, can also return the solution to x^2-ny^2 = +-1, +-2, +-3, and +-4)]

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[[Category:Diophantine equations]]
[[Category:Continued fractions]]