Pell's equation

As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the n = 2 case of Pell's equation,

$$x^2\; -\; 2\; y^2\; =\; 1,$$

and from the closely related equation

$$x^2\; -\; 2\; y^2\; =\; -1$$

because of the connection of these equations to the square root of 2. Indeed, if x and y are 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 side and diameter numbers, were known to the Pythagoreans, and Proclus observed that in the opposite direction these numbers obeyed one of these two equations. 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.{{MacTutor Biography|id=Baudhayana|title=Baudhayana}}

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.{{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 }}.

Likewise, Archimedes's cattle problem — an ancient word problem about finding the number of cattle belonging to the sun god Helios — 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,{{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 }} and the attribution to Archimedes is generally accepted today.{{cite book |title=Ptolemaic Alexandria |first=Peter M. |last=Fraser |author-link=Peter Fraser (classicist) |publisher=Oxford University Press |year=1972}}{{cite book |title=Number Theory, an Approach Through History |first=André |last=Weil |author-link=André Weil |publisher=Birkhäuser |year=1972}}

Around AD 250, Diophantus considered the equation

$$a^2\; x^2\; +\; c\; =\; y^2,$$

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.{{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}}

In Indian mathematics, Brahmagupta discovered that

$$(x\_1^2\; -\; Ny\_1^2)(x\_2^2\; -\; Ny\_2^2)\; =\; (x\_1x\_2\; +\; Ny\_1y\_2)^2\; -\; N(x\_1y\_2\; +\; x\_2y\_1)^2,$$

a form of what is now known as Brahmagupta's identity. Using this, he was able to "compose" triples $(x\_1,\; y\_1,\; k\_1)$ and $(x\_2,\; y\_2,\; k\_2)$ that were solutions of $x^2\; -\; Ny^2\; =\; k$, to generate the new triples

: $(x\_1x\_2\; +\; Ny\_1y\_2\; ,\; x\_1y\_2\; +\; x\_2y\_1\; ,\; k\_1k\_2)$ and $(x\_1x\_2\; -\; Ny\_1y\_2\; ,\; x\_1y\_2\; -\; x\_2y\_1\; ,\; k\_1k\_2).$

Not only did this give a way to generate infinitely many solutions to $x^2\; -\; Ny^2\; =\; 1$ starting with one solution, but also, by dividing such a composition by $k\_1k\_2$, integer or "nearly integer" solutions could often be obtained. For instance, for $N\; =\; 92$, Brahmagupta composed the triple (10, 1, 8) (since $10^2\; -\; 92(1^2)\; =\; 8$) with itself to get the new triple (192, 20, 64). Dividing throughout by 64 ("8" for $x$ and $y$) gave the triple (24, 5/2, 1), which when composed with itself gave the desired integer solution (1151, 120, 1). Brahmagupta solved many Pell's equations with this method, proving that it gives solutions starting from an integer solution of $x^2\; -\; Ny^2\; =\; k$ for k = ±1, ±2, or ±4.{{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}}.

The first general method for solving the Pell's equation (for all N) was given by Bhāskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method, it starts by choosing two relatively prime integers $a$ and $b$, then composing the triple $(a,\; b,\; k)$ (that is, one which satisfies $a^2\; -\; Nb^2\; =\; k$) with the trivial triple $(m,\; 1,\; m^2\; -\; N)$ to get the triple $\backslash big(am\; +\; Nb,\; a\; +\; bm,\; k(m^2\; -\; N)\backslash big)$, which can be scaled down to

$$\backslash left(\backslash frac\{am\; +\; Nb\}\{k\},\; \backslash frac\{a\; +\; bm\}\{k\},\; \backslash frac\{m^2\; -\; N\}\{k\}\backslash right).$$

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

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.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. 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=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}} 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 gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.In January 1658, at the end of {{lang|la|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 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 that Britain finally discarded in 1752: most of the rest of Europe would have regarded that date as January 31, 1658. See Old Style and New Style dates#Transposition of historical event dates and possible date conflicts.

John Pell's connection with the equation is that he revised Thomas Branker's translation{{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}}. of Johann Rahn's 1659 book Teutsche AlgebraTeutsch is an obsolete form of Deutsch, meaning "German". Free E-book: [https://books.google.com/books?id=ZJg_AAAAcAAJ Teutsche Algebra] at Google Books. 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.{{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}}

The general theory of Pell's equation, based on continued fractions and algebraic manipulations with numbers of the form $P\; +\; Q\backslash sqrt\{a\},$ was developed by Lagrange in 1766–1769."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. In particular, Lagrange gave a proof that the Brouncker–Wallis algorithm always terminates.

Let $h\_i/k\_i$ denote the unique sequence of convergents of the regular continued fraction for $\backslash sqrt\{n\}$. Then the pair of positive integers $(x\_1,\; y\_1)$ solving Pell's equation and minimizing x satisfies x₁ = h_i and y₁ = k_i for some i. This pair is called the fundamental solution. The sequence of integers $[a\_0;\; a\_1,a\_2,\backslash ldots]$ in the regular continued fraction of $\backslash sqrt\{n\}$ is always eventually periodic. It can be written in the form $\backslash left[\backslash lfloor\backslash sqrt\{n\}\backslash rfloor;\backslash ;\backslash overline\{a\_1,a\_2,\backslash ldots,a\_\{r-1\},\; 2\backslash lfloor\backslash sqrt\{n\}\backslash rfloor\}\backslash right]$, where $\backslash lfloor\backslash ,\; \backslash cdot\backslash ,\; \backslash rfloor$ denotes integer floor, and the sequence $a\_1,a\_2,\backslash ldots,a\_\{r-1\},\; 2\backslash lfloor\backslash sqrt\{n\}\backslash rfloor$ repeats infinitely. Moreover, the tuple $(a\_1,a\_2,\backslash ldots,a\_\{r-1\})$ is palindromic, the same left-to-right or right-to-left.

The fundamental solution is

(h_{2r-1},k_{2r-1}),&\text{ for }r\text{ odd}\end{cases}

The computation 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 $(x\_1,\; y\_1)$. 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.{{Citation |last=Lenstra |first=H. W. Jr. |title=Solving the Pell Equation |url=https://www.ams.org/notices/200202/fea-lenstra.pdf |journal=Notices of the American Mathematical Society |volume=49 |issue=2 |pages=182–192 |year=2002 |mr=1875156 |author-link=Hendrik Lenstra}}.

Once the fundamental solution is found, all remaining solutions may be calculated algebraically from

$$x\_k\; +\; y\_k\; \backslash sqrt\; n\; =\; (x\_1\; +\; y\_1\; \backslash sqrt\; n)^k,$$

expanding the right side, equating coefficients of $\backslash sqrt\{n\}$ on both sides, and equating the other terms on both sides. This yields the recurrence relations

$$x\_\{k+1\}\; =\; x\_1\; x\_k\; +\; n\; y\_1\; y\_k,$$

$$y\_\{k+1\}\; =\; x\_1\; y\_k\; +\; y\_1\; x\_k.$$

Although writing out the fundamental solution (x₁, y₁) 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

$$x\_1+y\_1\backslash sqrt\; n\; =\; \backslash prod\_\{i=1\}^t\; \backslash left(a\_i\; +\; b\_i\backslash sqrt\; n\backslash right)^\{c\_i\}$$

using much smaller integers a_i, b_i, and c_i.

For instance, Archimedes' cattle problem is equivalent to the Pell equation $x^2\; -\; 410\backslash ,286\backslash ,423\backslash ,278\backslash ,424\backslash \; y^2\; =\; 1$, the fundamental solution of which has {{val|206545}} digits if written out explicitly. However, the solution is also equal to

$$x\_1\; +\; y\_1\; \backslash sqrt\; n\; =\; u^\{2329\},$$

where

$$u\; =\; x\text{'}\_1\; +\; y\text{'}\_1\; \backslash sqrt\{4\backslash ,729\backslash ,494\}\; =\; (300\backslash ,426\backslash ,607\backslash ,914\backslash ,281\backslash ,713\backslash ,365\backslash \; \backslash sqrt\{609\}\; +\; 84\backslash ,129\backslash ,507\backslash ,677\backslash ,858\backslash ,393\backslash ,258\backslash \; \backslash sqrt\{7766\})^2$$

and $x\text{'}\_1$ and $y\text{'}\_1$ only have 45 and 41 decimal digits respectively.

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

$$\backslash exp\; O\backslash left(\backslash sqrt\{\backslash log\; N\backslash cdot\backslash log\backslash log\; N\}\backslash right),$$

where N = log n is the input size, similarly to the quadratic sieve.

Hallgren showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time.{{Citation |last=Hallgren |first=Sean |title=Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem |journal=Journal of the ACM |volume=54 |issue=1 |pages=1–19 |year=2007 |doi=10.1145/1206035.1206039 |s2cid=948064}}. 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 and Völlmer.{{Citation |last1=Schmidt |first1=A. |title=Proceedings of the thirty-seventh annual ACM symposium on Theory of computing – STOC '05 |pages=475–480 |year=2005 |chapter=Polynomial time quantum algorithm for the computation of the unit group of a number field |chapter-url=http://www.cdc.informatik.tu-darmstadt.de/reports/reports/Schmidt_Vollmer_STOC05.pdf |location=New York |publisher=ACM, Symposium on Theory of Computing |citeseerx=10.1.1.420.6344 |doi=10.1145/1060590.1060661 |isbn=1581139608 |last2=Völlmer |first2=U. |s2cid=6654142 |access-date=25 October 2017 |archive-date=26 October 2017 |archive-url=https://web.archive.org/web/20171026110757/https://www.cdc.informatik.tu-darmstadt.de/reports/reports/Schmidt_Vollmer_STOC05.pdf |url-status=dead }}.

As an example, consider the instance of Pell's equation for n = 7; that is,

$$x^2\; -\; 7y^2\; =\; 1.$$

The continued fraction of $\backslash sqrt\{7\}$ has the form $[2;\backslash \; \backslash overline\{1,\; 1,\; 1,\; 4\}]$. Since the period has length $4$, which is an even number, the convergent producing the fundamental solution is obtained by truncating the continued fraction right before the end of the first occurrence of the period: $[2;\backslash \; 1,1,1]=\backslash frac\{8\}\{3\}$.

The sequence of convergents for the square root of seven are

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)

For the Pell's equation

$$x^2\; -\; 13y^2\; =\; 1,$$

the continued fraction $\backslash sqrt\{13\}=[3;\backslash \; \backslash overline\{1,1,1,1,6\}]$ has a period of odd length. For this the fundamental solution is obtained by truncating the continued fraction right before the second occurrence of the period $[3;\backslash \; 1,1,1,1,6,1,1,1,1]=\backslash frac\{649\}\{180\}$. Thus, the fundamental solution is $(x\_1,\; y\_1)=(649,\; 180)$.

The smallest solution can be very large. For example, the smallest solution to $x^2\; -\; 313y^2\; =\; 1$ is ({{val|32188120829134849}}, {{val|1819380158564160}}), and this is the equation which Frenicle challenged Wallis to solve.{{cite web |url = http://primes.utm.edu/curios/page.php?short=313 |title = Prime Curios!: 313}} Values of n such that the smallest solution of $x^2\; -\; ny^2\; =\; 1$ 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}} for x and {{oeis|id=A033319}} for y.)

The following is a list of the fundamental solution to $x^2\; -\; n\; y^2\; =\; 1$ with n ≤ 128. When n is an integer square, there is no solution except for the trivial solution (1, 0). The values of x are sequence {{OEIS link|id=A002350}} and those of y are sequence {{OEIS link|id=A002349}} in OEIS.

Pell's equation has connections to several other important subjects in mathematics.

Pell's equation is closely related to the theory of algebraic numbers, as the formula

$$x^2\; -\; n\; y^2\; =\; (x\; +\; y\backslash sqrt\; n)(x\; -\; y\backslash sqrt\; n)$$

is the norm for the ring $\backslash mathbb\{Z\}[\backslash sqrt\{n\}]$ and for the closely related quadratic field $\backslash mathbb\{Q\}(\backslash sqrt\{n\})$. Thus, a pair of integers $(x,\; y)$ solves Pell's equation if and only if $x\; +\; y\; \backslash sqrt\{n\}$ is a unit with norm 1 in $\backslash mathbb\{Z\}[\backslash sqrt\{n\}]$.{{Cite journal |last=Clark |first=Pete |title=Number Theory: A Contemporary Introduction |url=http://alpha.math.uga.edu/~pete/4400FULL2018.pdf|journal=University of Georgia}} Ch. 7 The Pell Equation Dirichlet's unit theorem, that all units of $\backslash mathbb\{Z\}[\backslash sqrt\{n\}]$ can be expressed as powers of a single fundamental unit (and multiplication by a sign), is an algebraic restatement of the fact that all solutions to the Pell's equation can be generated from the fundamental solution.{{Cite web |last=Conrad |first=Keith |title=Dirichlet's Unit Theorem |url=https://kconrad.math.uconn.edu/blurbs/gradnumthy/unittheorem.pdf |access-date=14 July 2020}} 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.

Demeyer mentions a connection between Pell's equation and the Chebyshev polynomials:

If $T\_i(x)$ and $U\_i(x)$ 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 $n\; =\; x^2\; -\; 1$:{{Citation |last=Demeyer |first=Jeroen |title=Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields |url=http://cage.ugent.be/~jdemeyer/phd.pdf |page=70 |year=2007 |archive-url=https://web.archive.org/web/20070702185523/https://cage.ugent.be/~jdemeyer/phd.pdf |series=PhD thesis, Ghent University |access-date=27 February 2009 |archive-date=2 July 2007 |url-status=dead}}.

$$T\_i^2\; -\; (x^2-1)\; U\_\{i-1\}^2\; =\; 1.$$

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

$$T\_i\; +\; U\_\{i-1\}\; \backslash sqrt\{x^2-1\}\; =\; (x\; +\; \backslash sqrt\{x^2-1\})^i.$$

It may further be observed that if $(x\_i,\; y\_i)$ are the solutions to any integer Pell's equation, then $x\_i\; =\; T\_i\; (x\_1)$ and $y\_i\; =\; y\_1\; U\_\{i-1\}\; (x\_1)$.{{Citation |last=Barbeau |first=Edward J. |title=Pell's Equation |url=https://archive.org/details/pellsequation0000barb |chapter=3. Quadratic surds |year=2003 |series=Problem Books in Mathematics |publisher=Springer-Verlag |isbn=0-387-95529-1 |mr=1949691 |url-access=registration}}.

A general development of solutions of Pell's equation $x^2\; -\; n\; y^2\; =\; 1$ in terms of continued fractions of $\backslash sqrt\{n\}$ 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 irrationals.

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

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_k−1/q_k−1 and p_k/q_k are two successive convergents of a continued fraction, then the matrix

has determinant (−1)^k.

Størmer's theorem applies Pell equations to find pairs of consecutive smooth numbers, positive integers whose prime factors are all smaller than a given value.{{cite journal |last=Størmer |first=Carl |author-link=Carl Størmer |title=Quelques théorèmes sur l'équation de Pell $x^2\; -\; Dy^2\; =\; \backslash pm1$ et leurs applications |language=fr |journal=Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. |volume=I |issue=2 |year=1897}}{{cite journal |last=Lehmer |first=D. H. |author-link=D. H. Lehmer |year=1964 |title=On a Problem of Størmer |journal=Illinois Journal of Mathematics |volume=8 |issue=1 |pages=57–79 |mr=0158849 |doi=10.1215/ijm/1256067456 |doi-access=free}} As part of this theory, 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 n.

The negative Pell's equation is given by

$$x^2\; -\; n\; y^2\; =\; -1$$

and has also been extensively studied. It can be solved by the same method of continued fractions and has solutions if and only if the period of the continued fraction has odd length. A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3.This is because the Pell equation implies that −1 is a quadratic residue modulo n. Thus, for example, x² − 3 y² = −1 is never solvable, but x² − 5 y² = −1 may be.{{Cite journal |last1=Wang |first1=Jiaqi |last2=Cai |first2=Lide |date=December 2013 |title=The solvability of negative Pell equation |url=http://archive.ymsc.tsinghua.edu.cn/pacm_download/21/241-2013The_solvability_of_negative_Pell_equation.pdf |journal=Tsinghua College |pages=5–6}}

The first few numbers n for which x² − n y² = −1 is solvable are 1 (with only one trivial solution) and

:2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, ... {{OEIS|id=A031396}}

with infinitely many solutions. The solutions of the negative Pell's equation for $1\; \backslash le\; n\; \backslash le\; 298$ are:

Let $\backslash alpha\; =\; \backslash Pi\_\{j\backslash text\{\; is\; odd\}\}\; (1\; -\; 2^j)$. The proportion of square-free n divisible by k primes of the form 4m + 1 for which the negative Pell's equation is solvable is at least α.{{Citation |last1=Cremona |first1=John E. |title=Some density results for negative Pell equations; an application of graph theory |journal=Journal of the London Mathematical Society |volume=39 |issue=1 |pages=16–28 |year=1989 |series=Second Series |doi=10.1112/jlms/s2-39.1.16 |issn=0024-6107 |last2=Odoni |first2=R. W. K.}}. When the number of prime divisors is not fixed, the proportion is given by 1 − α.{{Cite web |date=2022-08-10 |title=Ancient Equations Offer New Look at Number Groups |url=https://www.quantamagazine.org/ancient-equations-offer-new-look-at-number-groups-20220810/ |access-date=2022-08-18 |website=Quanta Magazine |language=en}}{{cite arXiv |last1=Koymans |first1=Peter |last2=Pagano |first2=Carlo |date=2022-01-31 |title=On Stevenhagen's conjecture |class=math.NT |eprint=2201.13424 }}

If the negative Pell's 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:

$$(x^2\; -\; n\; y^2)^2\; =\; (-1)^2$$

implies

$$>(x^2\; +\; n\; y^2)^2\; -\; n(2xy)^2\; =\; 1.$$

As stated above, if the negative Pell's equation is solvable, a solution can be found using the method of continued fractions as in the positive Pell's equation. The recursion relation works slightly differently however. Since $(x\; +\; y\backslash sqrt\{n\})(x\; -\; y\backslash sqrt\{n\})\; =\; -1$, the next solution is determined in terms of $i(x\_k\; +\; y\_k\backslash sqrt\{n\})\; =\; (i(x\; +\; y\backslash sqrt\{n\}))^k$ whenever there is a match, that is, when $k$ is odd. The resulting recursion relation is (modulo a minus sign, which is immaterial due to the quadratic nature of the equation)

$$x\_k\; =\; x\_\{k-2\}\; x\_1^2\; +\; n\; x\_\{k-2\}\; y\_1^2\; +\; 2\; n\; y\_\{k-2\}\; y\_1\; x\_1,$$

$$y\_k\; =\; y\_\{k-2\}\; x\_1^2\; +\; n\; y\_\{k-2\}\; y\_1^2\; +\; 2\; x\_\{k-2\}\; y\_1\; x\_1,$$

which gives an infinite tower of solutions to the negative Pell's equation (except for $n\; =\; 1$).

The equation

$$x^2\; -\; n\; y^2\; =\; N$$

is called the generalized{{cite book |last=Peker |first=Bilge |date=2021 |title=Current Studies in Basic Sciences, Engineering and Technology 2021 |url=https://www.isres.org/books/chapters/CSBET2021_10_03-01-2022.pdf |publisher= ISRES Publishing |page=136 |access-date=2024-02-25}}{{cite report |last=Tamang |first=Bal Bahadur |date=August 2022 |title=Solvability of Generalized Pell's Equation and Its Applications in Real Life |url=https://www.researchgate.net/publication/363254319 |publisher=Tribhuvan University |access-date=2024-02-25}} (or general{{Cite book |last1=Andreescu |first1=Titu |title=Quadratic Diophantine Equations |last2=Andrica |first2=Dorin |publisher=Springer |year=2015 |isbn=978-0-387-35156-8 |location=New York}}) Pell's equation. The equation $u^2\; -\; n\; v^2\; =\; 1$ is the corresponding Pell's resolvent. A recursive algorithm was given by Lagrange in 1768 for solving the equation, reducing the problem to the case .{{Cite book |last=Lagrange |first=Joseph-Louis |url=https://gallica.bnf.fr/ark:/12148/bpt6k215570z |title=Oeuvres de Lagrange. T. 2 / publiées par les soins de M. J.-A. Serret [et G. Darboux]; [précédé d'une notice sur la vie et les ouvrages de J.-L. Lagrange, par M. Delambre] |date=1867–1892 |language=fr}}{{Cite web |last=Matthews |first=Keith |title=The Diophantine Equation x² − Dy² = N, D > 0 |url=http://www.numbertheory.org/pdfs/patz5.pdf |archive-url=https://web.archive.org/web/20150318090657/http://www.numbertheory.org/pdfs/patz5.pdf |archive-date=2015-03-18 |url-status=live |access-date=20 July 2020}} Such solutions can be derived using the continued-fractions method as outlined above.

If $(x\_0,\; y\_0)$ is a solution to $x^2\; -\; n\; y^2\; =\; N,$ and $(u\_k,\; v\_k)$ is a solution to $u^2\; -\; n\; v^2\; =\; 1,$ then $(x\_k,\; y\_k)$ such that $x\_k\; +\; y\_k\; \backslash sqrt\{n\}\; =\; \backslash big(x\_0\; +\; y\_0\; \backslash sqrt\{n\}\backslash big)\backslash big(u\_k\; +\; v\_k\; \backslash sqrt\{n\}\backslash big)$ is a solution to $x^2\; -\; n\; y^2\; =\; N$, a principle named the multiplicative principle. The solution $(x\_k,\; y\_k)$ is called a Pell multiple of the solution $(x\_0,\; y\_0)$.

There exists a finite set of solutions to $x^2\; -\; n\; y^2\; =\; N$ such that every solution is a Pell multiple of a solution from that set. In particular, if $(u,\; v)$ is the fundamental solution to $u^2\; -\; n\; v^2\; =\; 1$, then each solution to the equation is a Pell multiple of a solution $(x,\; y)$ with $|x|\; \backslash le\; \backslash tfrac\{1\}\{2\}\; \backslash sqrt$

If x and y are positive integer solutions to the Pell's equation with , then $x/y$ is a convergent to the continued fraction of $\backslash sqrt\; n$.

Solutions to the generalized Pell's equation are used for solving certain Diophantine equations and units of certain rings,{{Cite journal |last=Bernstein |first=Leon |date=1975-10-01 |title=Truncated units in infinitely many algebraic number fields of degreen ≧4 |journal=Mathematische Annalen |language=en |volume=213 |issue=3 |pages=275–279 |doi=10.1007/BF01350876 |s2cid=121165073 |issn=1432-1807}}{{Cite journal |last=Bernstein |first=Leon |date=1 March 1974 |title=On the Diophantine Equation x(x + n)(x + 2n) + y(y + n)(y + 2n) = z(z + n)(z + 2n) |journal=Canadian Mathematical Bulletin |language=en |volume=17 |issue=1 |pages=27–34 |doi=10.4153/CMB-1974-005-5 |s2cid=125002637 |issn=0008-4395|doi-access=free }} and they arise in the study of SIC-POVMs in quantum information theory.{{Cite journal |last1=Appleby |first1=Marcus |last2=Flammia |first2=Steven |last3=McConnell |first3=Gary |last4=Yard |first4=Jon |date=August 2017 |title=SICs and Algebraic Number Theory |journal=Foundations of Physics |language=en |volume=47 |issue=8 |pages=1042–1059 |arxiv=1701.05200 |bibcode=2017FoPh...47.1042A |doi=10.1007/s10701-017-0090-7 |s2cid=119334103 |issn=0015-9018}}

The equation

$$x^2\; -\; n\; y^2\; =\; 4$$

is similar to the resolvent $x^2\; -\; n\; y^2\; =\; 1$ in that if a minimal solution to $x^2\; -\; n\; y^2\; =\; 4$ can be found, then all solutions of the equation can be generated in a similar manner to the case $N\; =\; 1$. For certain $n$, solutions to $x^2\; -\; n\; y^2\; =\; 1$ can be generated from those with $x^2\; -\; n\; y^2\; =\; 4$, in that if $n\; \backslash equiv\; 5\; \backslash pmod\{8\},$ then every third solution to $x^2\; -\; n\; y^2\; =\; 4$ has $x,\; y$ even, generating a solution to $x^2\; -\; n\; y^2\; =\; 1$.

{{reflist|group=note}}

• {{Cite book |last=Edwards |first=Harold M. |title=Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory |publisher=Springer-Verlag |year=1996 |isbn=0-387-90230-9 |series=Graduate Texts in Mathematics |volume=50 |mr=0616635 |ref=none |author-link=Harold Edwards (mathematician) |orig-year=1977}}
• {{Cite journal |last=Pinch |first=R. G. E. |year=1988 |title=Simultaneous Pellian equations |url=http://www.s369624816.websitehome.co.uk/rgep/p08.ps |volume=103 |issue=1 |pages=35–46 |bibcode=1988MPCPS.103...35P |doi=10.1017/S0305004100064598 |ref=none |journal=Mathematical Proceedings of the Cambridge Philosophical Society |s2cid=123098216 }}
• {{Cite book |last=Whitford |first=Edward Everett |year=1912 |title=The Pell equation |url=http://name.umdl.umich.edu/ABV2773.0001.001 |publisher=Columbia University |format=PhD Thesis |ref=none}}
• {{Cite book |last=Williams |first=H. C. |year=2002 |editor-last=Bennett |editor-first=M. A. |title=Surveys in number theory: Papers from the millennial conference on number theory |chapter=Solving the Pell equation |location=Natick, MA |publisher=A K Peters |pages=325–363 |isbn=1-56881-162-4 |zbl=1043.11027 |ref=none |editor-last2=Berndt |editor-first2=B. C. |editor-link2=Bruce C. Berndt |editor-last3=Boston |editor-first3=N. |editor-link3=Nigel Boston |editor-last4=Diamond |editor-first4=H. G. |editor-last5=Hildebrand |editor-first5=A. J. |editor-last6=Philipp |editor-first6=W.}}
• {{cite book | last=Andreescu | first=Titu | last2=Andrica | first2=Dorin | title=Quadratic Diophantine Equations | publisher=Springer | publication-place=New York | date=2015 | isbn=978-0-387-35156-8 | oclc=916486370}}
• {{cite book | last=Jacobson | first=Michael | last2=Williams | first2=Hugh | title=Solving the Pell Equation | publisher=Springer-Verlag | publication-place=New York, NY | date=2008 | isbn=978-0-387-84922-5 | oclc=245561348 }}

• {{mathworld|urlname=PellEquation|title=Pell's equation|ref=none}}
• {{MacTutor|class=HistTopics|id=Pell|title=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 < 10¹⁰, can also return the solution to x² − ny² = ±1, ±2, ±3, and ±4)

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