Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form

x^2 - ny^2 = 1,

where ''n'' is a given positive nonsquare

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...

; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the

trivial solution In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...

with ''x'' = 1 and ''y'' = 0.

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaperfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately

approximate An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...

the

square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...

of ''n'' by

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

s of the form ''x''/''y''. This equation was first studied extensively

in India IN, In or in may refer to: Places * India (country code IN) * Indiana, United States (postal code IN) * Ingolstadt, Germany (license plate code IN) * In, Russia, a town in the Jewish Autonomous Oblast Businesses and organizations * Indepen ...

starting with

Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical trea ...

, who found an integer solution to

92x^2 + 1 = y^2

in his ''

Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good unders ...

'' circa 628. Bhaskara II in the 12th century and

Narayana Pandit Nārāyaṇa Paṇḍita ( sa, नारायण पण्डित) (1340–1400) was an Indian mathematician. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the K ...

in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Bhaskara II is generally credited with developing the ''chakravala'' method, building on the work of Jayadeva and Brahmagupta. Solutions to specific examples of Pell's equation, such as the

Pell number In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins , , , , an ...

s arising from the equation with ''n'' = 2, had been known for much longer, since the time of

Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...

Greece Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders with ...

and a similar date in India. William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose from

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

mistakenly attributing Brouncker's solution of the equation to John Pell.In Euler's (pp. 227ff), he presents a solution to Pell's equation which was taken from John Wallis' , specifically, Letter 17 () and Letter 19 () of: * The letters are in Latin. Letter 17 appears on pp. 56–72. Letter 19 appears on pp. 81–91. * French translations of Wallis' letters: 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' (1693), which includes articles by John Pell: * 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: :* (On Algebra by Dr. John Pell and especially on an incompletely determined problem), pp. 234–236. :* (Example of Pell's method), pp. 238–244. :* (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)
p. 52
*

History

As early as 400 BC in

India India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...

and

, mathematicians studied the numbers arising from the ''n'' = 2 case of Pell's equation, :

x^2 - 2y^2 = 1,

and from the closely related equation :

x^2 - 2y^2 = -1

because of the connection of these equations to the

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...

. Indeed, if ''x'' and ''y'' are

positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...

satisfying this equation, then ''x''/''y'' is an approximation of . The numbers ''x'' and ''y'' appearing in these approximations, called side and diameter numbers, were known to the

Pythagoreans Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in the ancient Greek colony of Kroton, ...

, and

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor ( grc-gre, Πρόκλος ὁ Διάδοχος, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers ...

observed that in the opposite direction these numbers obeyed one of these two equations. Similarly,

Baudhayana The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...

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 Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

approximated the

square root of 3 The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative nu ...

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.. Likewise,

Archimedes's cattle problem Archimedes's cattle problem (or the or ) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from ...

— an ancient word problem about finding the number of cattle belonging to the sun god

Helios In ancient Greek religion and Greek mythology, mythology, Helios (; grc, , , Sun; Homeric Greek: ) is the deity, god and personification of the Sun (Solar deity). His name is also Latinized as Helius, and he is often given the epithets Hyper ...

— 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 Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...

, and the attribution to Archimedes is generally accepted today. Around AD 250,

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

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. In Indian mathematics,

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 In algebra, Brahmagupta's identity says that, for given n, the product of two numbers of the form a^2+nb^2 is itself a number of that form. In other words, the set of such numbers is closed under multiplication. Specifically: :\begin \left(a^2 + n ...

. 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.. 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

\big(am + Nb, a + bm, k(m^2 - N)\big)

, which can be scaled down to :

\left(\frac, \frac, \frac\right).

When

m

is chosen so that

\frac

is an integer, so are the other two numbers in the triple. Among such

m

, the method chooses one that minimizes

\frac

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'' = , ''y'' = to the ''N'' = 61 case. Several European mathematicians rediscovered how to solve Pell's equation in the 17th century.

Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to

Kenelm Digby Sir Kenelm Digby (11 July 1603 – 11 June 1665) was an English courtier and diplomat. He was also a highly reputed natural philosopher, astrologer and known as a leading Roman Catholic intellectual and Blackloist. For his versatility, he is d ...

Bernard Frénicle de Bessy Bernard Frénicle de Bessy (c. 1604 – 1674), was a French mathematician born in Paris, who wrote numerous mathematical papers, mainly in number theory and combinatorics. He is best remembered for , a treatise on magic squares published posth ...

said that Fermat found the smallest solution for ''N'' up to 150 and challenged

John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...

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. John Pell's connection with the equation is that he revised

Thomas Branker Thomas Branker (Brancker) (1633–1676) was an English mathematician. Life He was born at Barnstaple in August 1633, the son of another Thomas Brancker, a graduate of Exeter College, Oxford, who was in 1626 a schoolmaster near Ilchester, and abou ...

's translation of

Johann Rahn Johann Rahn (Latinised form Rhonius) (10 March 1622 – 25 May 1676) was a Swiss mathematician who is credited with the first use of the division sign, ÷ (a repurposed obelus variant) and the therefore sign, ∴. The symbols were used in ''Teu ...

's 1659 book ''Teutsche Algebra'''' Teutsch'' is an obsolete form of ''Deutsch'', meaning "German". Free E-book
''Teutsche Algebra''
at Google Books. into English, with a discussion of Brouncker's solution of the equation.

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 In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

s and algebraic manipulations with numbers of the form

P + Q\sqrt,

was developed by Lagrange in 1766–1769.

Solutions

Fundamental solution via continued fractions

Let

h_i/k_i

denote the sequence of convergents to the regular continued fraction for

\sqrt

. This sequence is unique. Then the pair

(x_1, y_1)

solving Pell's equation and minimizing ''x'' satisfies ''x''₁ = ''h_i'' and ''y''₁ = ''k_i'' 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. The time for finding the fundamental solution using the continued fraction method, with the aid of the

Schönhage–Strassen algorithm The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers. It was developed by Arnold Schönhage and Volker Strassen in 1971.A. Schönhage and V. Strassen,Schnelle Multiplikation großer Zahlen, '' ...

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 In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...

because the number of digits in the solution may be as large as , far larger than a polynomial in the number of digits in the input value ''n''..

Additional solutions from the fundamental solution

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

x_k + y_k \sqrt n = (x_1 + y_1 \sqrt n)^k,

expanding the right side,

equating coefficients In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when co ...

\sqrt

on both sides, and equating the other terms on both sides. This yields the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s :

x_ = x_1 x_k + n y_1 y_k,

y_ = x_1 y_k + y_1 x_k.

Concise representation and faster algorithms

Although writing out the fundamental solution (''x''₁, ''y''₁) 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\sqrt n = \prod_^t (a_i + b_i\sqrt n)^

using much smaller integers ''a''_''i'', ''b''_''i'', and ''c''_''i''. For instance,

Archimedes' cattle problem Archimedes's cattle problem (or the or ) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from ...

is equivalent to the Pell equation

x^2 - 410\,286\,423\,278\,424 y^2 = 1

, the fundamental solution of which has digits if written out explicitly. However, the solution is also equal to :

x_1 + y_1 \sqrt n = u^,

where :

u = x'_1 + y'_1 \sqrt = (300\,426\,607\,914\,281\,713\,365 \sqrt + 84\,129\,507\,677\,858\,393\,258 \sqrt)^2

and

x'_1

and

y'_1

only have 45 and 41 decimal digits respectively. Methods related to the

quadratic sieve The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerab ...

approach for

integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

may be used to collect relations between prime numbers in the number field generated by 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 The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...

, it can be shown to take time :

\exp O(\sqrt),

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

Quantum algorithms

Hallgren showed that a

quantum computer Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...

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 In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...

, was extended to more general fields by Schmidt and Völlmer.

Example

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

x^2 - 7y^2 = 1.

The sequence of convergents for the square root of seven are : 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 (''x'') and (''y'') in

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

) The smallest solution can be very large. For example, the smallest solution to

x^2 - 313y^2 = 1

is (, ), and this is the equation which Frenicle challenged Wallis to solve. 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, ... . (For these records, see for ''x'' and for ''y''.)

List of fundamental solutions of Pell's equations

The following is a list of the fundamental solution to

x^2 - ny^2 = 1

with ''n'' ≤ 128. For square ''n'', there is no solution except (1, 0). The values of ''x'' are sequence and those of ''y'' are sequence in

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 An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...

s, as the formula :

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

is the

norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...

for the

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

\mathbb

sqrt In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...

/math> and for the closely related

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...

\mathbb(\sqrt)

. Thus, a pair of integers

(x, y)

solves Pell's equation if and only if

x + y \sqrt

is a

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

with norm 1 in

\mathbb

/math>.

Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...

, that all units of

\mathbb

/math> 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. 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 mentions a connection between Pell's equation and the

Chebyshev polynomials The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as T_n(x) and U_n(x). They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshe ...

: 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 In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...

R /math>, with n = x^2 - 1 :

: T_i^2 - (x^2-1) U_^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_ \sqrt = (x + \sqrt)^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_ (x_1) .

Continued fractions

A general development of solutions of Pell's equation

x^2 - ny^2 = 1

in terms of

s of

\sqrt

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 In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducibl ...

s. The relationship to the continued fractions implies that the solutions to Pell's equation form a

semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...

subset of the modular group. Thus, for example, if ''p'' and ''q'' satisfy Pell's equation, then :

\begin p & q \\ nq & p \end

is a matrix of unit

determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...

. 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 :

\begin p_ & p_ \\ q_ & q_ \end

has determinant (−1)^''k''.

Smooth numbers

Størmer's theorem applies Pell equations to find pairs of consecutive

smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 � ...

s, positive integers whose prime factors are all smaller than a given value. 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

that does not divide ''n''.

The negative Pell's equation

The negative Pell's equation is given by :

x^2 - ny^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. 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'' + 3.This is because the Pell equation implies that −1 is a quadratic residue modulo ''n''. Thus, for example, ''x''² − 3''ny''² = −1 is never solvable, but ''x''² − 5''ny''² = −1 may be. The first few numbers ''n'' for which ''x''² − ''ny''² = −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, ... . Let

\alpha = \Pi_ (1 - 2^j)

. The proportion of square-free ''n'' divisible by ''k'' primes of the form 4''m'' + 1 for which the negative Pell's equation is solvable is at least ''α''. When the number of prime divisors is not fixed, the proportion is given by 1 - ''α.'' 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 - ny^2)^2 = (-1)^2

implies :

(x^2 + ny^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 + \sqrt y)(x - \sqrt y) = -1

, the next solution is determined in terms of

i(x_k + \sqrt y_k) = (i(x + \sqrt y))^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_ x_1^2 + n x_ y_1^2 + 2 n y_ y_1 x_1,

y_k = y_ x_1^2 + n y_ y_1^2 + 2 x_ y_1 x_1,

which gives an infinite tower of solutions to the negative Pell's equation.

Generalized Pell's equation

The equation :

x^2 - dy^2 = N

is called the generalized (or general) Pell's equation. The equation

u^2 - dv^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

, N,  < \sqrt

. Such solutions can be derived using the continued-fractions method as outlined above. If

(x_0, y_0)

is a solution to

x^2 - dy^2 = N,

and

(u_n, v_n)

is a solution to

u^2 - dv^2 = 1,

then

(x_n, y_n)

such that

x_n + y_n \sqrt = \big(x_0 + y_0 \sqrt\big)\big(u_n + v_n \sqrt\big)

is a solution to

x^2 - dy^2 = N

, a principle named the ''multiplicative principle''. The solution

(x_n, y_n)

is called a ''Pell multiple'' of the solution

(x_0, y_0)

. There exists a finite set of solutions to

x^2 - dy^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 - dv^2 = 1

, then each solution to the equation is a Pell multiple of a solution

(x, y)

with

, x,  \le \sqrt \left(\sqrt + 1\right) / 2

and

, y,  \le \sqrt \left(\sqrt + 1\right) / \big(2\sqrt\big)

, where

U = u + v \sqrt d

. If ''x'' and ''y'' are positive integer solutions to the Pell's equation with

, N,  < \sqrt d

, then

x/y

is a convergent to the continued fraction of

\sqrt d

. Solutions to the generalized Pell's equation are used for solving certain Diophantine equations and

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. The equation :

x^2 - dy^2 = 4

is similar to the resolvent

x^2 - dy^2 = 1

in that if a minimal solution to

x^2 - dy^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

d

, solutions to

x^2 - dy^2 = 1

can be generated from those with

x^2 - dy^2 = 4

, in that if

d \equiv 5 \pmod,

then every third solution to

x^2 - dy^2 = 4

has

x, y

even, generating a solution to

x^2 - dy^2 = 1

Notes

References

External links

* *
Pell equation solver
(''n'' has no upper limit)

(''n'' < 10¹⁰, can also return the solution to ''x''² − ''ny''² = ±1, ±2, ±3, and ±4) {{DEFAULTSORT:Pell's Equation Diophantine equations Continued fractions

History

Solutions

Fundamental solution via continued fractions

Additional solutions from the fundamental solution

Concise representation and faster algorithms

Quantum algorithms

Example

List of fundamental solutions of Pell's equations

Connections

Algebraic number theory

Chebyshev polynomials

Continued fractions

Smooth numbers

The negative Pell's equation

Generalized Pell's equation

Notes

References

Further reading

External links