Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form
where n is a given positive 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 trivial solution with x = 1 and y = 0. Joseph Louis Lagrange proved that, as long as n is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y.
This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to in his Brāhmasphuṭasiddhānta circa 628. Bhaskara II in the twelfth century and Narayana Pandit in the fourteenth 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 numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and 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.[note 1]
As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the n = 2 case of Pell's equation,
and from the closely related equation
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 √. 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.
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. 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, and the attribution to Archimedes is generally accepted today.
Around AD 250, Diophantus considered the equation
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
a form of what is now known as Brahmagupta's identity. Using this, he was able to "compose" triples and that were solutions of , to generate the new triples
Not only did this give a way to generate infinitely many solutions to starting with one solution, but also, by dividing such a composition by , integer or "nearly integer" solutions could often be obtained. For instance, for , Brahmagupta composed the triple (10, 1, 8) (since ) with itself to get the new triple (192, 20, 64). Dividing throughout by 64 ('8' for and ) 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, proving that it gives solutions starting from an integer solution of for k = ±1, ±2, or ±4.
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 (cyclic) method, it starts by choosing two relatively prime integers and , then composing the triple (that is, one which satisfies ) with the trivial triple to get the triple , which can be scaled down to
When is chosen so that is an integer, so are the other two numbers in the triple. Among such , the method chooses one that minimizes , 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 = 226153980 to the N = 61 case.
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. 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.
John Pell's connection with the equation is that he revised Thomas Branker's translation of Johann Rahn's 1659 book Teutsche Algebra[note 2] 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.
Fundamental solution via continued fractions
Let denote the sequence of convergents to the regular continued fraction for . This sequence is unique. Then the pair (x1,y1) solving Pell's equation and minimizing x satisfies x1 = hi and y1 = ki 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 for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair (x1,y1). However, this is not a polynomial time algorithm 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
Concise representation and faster algorithms
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
using much smaller integers ai, bi, and ci.
For instance, Archimedes' cattle problem is equivalent to the Pell equation , the fundamental solution of which has 206545 digits if written out explicitly. However, the solution is also equal to
and and 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 √, 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
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. 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.
As an example, consider the instance of Pell's equation for n = 7; that is,
The sequence of convergents for the square root of seven are
h / k (Convergent) h2 − 7k2 (Pell-type approximation) 2 / 1 −3 3 / 1 +2 5 / 2 −3 8 / 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 A001081 (x) and A001080 (y) in OEIS)
The smallest solution can be very large. For example, the smallest solution to is (32188120829134849, 1819380158564160), and this is the equation which Frenicle challenged Wallis to solve. Values of n such that the smallest solution of 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, ... (sequence A033316 in the OEIS).
The smallest solution of Pell equations
The following is a list of the smallest solution (fundamental solution) to with n ≤ 128. For square n, there is no solution except (1, 0). The values of x are sequence A002350 and those of y are sequence A002349 in OEIS.
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 numbers, as the formula
is the norm for the ring and for the closely related quadratic field . Thus, a pair of integers solves Pell's equation if and only if is a unit with norm 1 in . Dirichlet's unit theorem, that all units of 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 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.
Demeyer mentions a connection between Pell's equation and the Chebyshev polynomials: If Ti (x) and Ui (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 = x2 − 1:
Thus, these polynomials can be generated by the standard technique for Pell equations of taking powers of a fundamental solution:
It may further be observed that, if (xi,yi) are the solutions to any integer Pell equation, then xi = Ti (x1) and yi = y1Ui − 1(x1).
A general development of solutions of Pell's equation in terms of continued fractions of 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.
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 pk−1/qk−1 and pk/qk 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. 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 equation
The negative Pell equation is given by
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 4k + 3.[note 3] Thus, for example, x2 − 3ny2 = −1 is never solvable, but x2 − 5ny2 = −1 may be.
The first few numbers n for which x2 − ny2 = −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, ... (sequence A031396 in the OEIS).
The proportion of square-free n divisible by k primes of the form 4m + 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:
Generalized Pell's equation
is called the generalized (or general) Pell's equation. The equation 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 . Such solutions can be derived using the continued fractions method as outlined above.
If is a solution to and is a solution to then such that is a solution to , a principle named the multiplicative principle.
Solutions to the generalized Pell's equation are used for solving certain Diophantine equations and units of certain rings, and they arise in the study of SIC-POVMs in quantum information theory.
is similar to the resolvent in that if a minimal solution to can be found then all solutions of the equation can be generated in a similar manner to the case . For certain , solutions to can be generated from those with , in that if then every third solution to has x,y even, generating a solution to .
- 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:
- Wallis, John, ed. (1658). Commercium epistolicum, de Quaestionibus quibusdam Mathematicis nuper habitum [Correspondence, about some mathematical inquiries recently undertaken] (in English, Latin, and French). Oxford, England: A. Lichfield. The letters are in Latin. Letter 17 appears on pp. 56–72. Letter 19 appears on pp. 81–91.
- French translations of Wallis' letters: Fermat, Pierre de (1896). Tannery, Paul; Henry, Charles (eds.). Oeuvres de Fermat (in French and Latin). 3rd vol. Paris, France: Gauthier-Villars et fils. Letter 17 appears on pp. 457–480. Letter 19 appears on pp. 490–503.
- Wallis, John (1693). Opera mathematica: de Algebra Tractatus; Historicus & Practicus [Mathematical works: Treatise on Algebra; historical and as presently practiced] (in Latin, English, and French). 2nd vol. Oxford, England. 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.
- Whitford, Edward Everett (1912) "The Pell equation," doctoral thesis, Columbia University (New York, New York, USA), p. 52.
- Heath, Thomas L. (1910). Diophantus of Alexandria : A Study in the History of Greek Algebra. Cambridge, England: Cambridge University Press. p. 286.
- Teutsch is an obsolete form of Deutsch, meaning "German". Free E-book: Teutsche Algebra (Google Books)
- This is because the Pell equation implies that −1 is a quadratic residue modulo n.
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- 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):
- Euler, Leonhard (1732–1733). "De solutione problematum Diophantaeorum per numeros integros" [On the solution of Diophantine problems by integers]. Commentarii Academiae Scientiarum Imperialis Petropolitanae (Memoirs of the Imperial Academy of Sciences at St. Petersburg). 6: 175–188. 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: Fuss, P.H., ed. (1843). Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … [Mathematical and physical correspondence of some famous geometers of the 18th century …] (in French, Latin, and German). St. Petersburg, Russia. p. 37. 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.)
- Euler, Leonhard (1771). Vollständige Anleitung zur Algebra, II. Theil [Complete Introduction to Algebra, Part 2] (in German). Kayserlichen Akademie der Wissenschaften (Imperial Academy of Sciences): St. Petersburg, Russia. p. 227. 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: Euler, Leonhard (1810). Elements of Algebra …. 2nd vol. (2nd ed.). London, England: J. Johnson. p. 78.
- Heath, Thomas L. (1910). Diophantus of Alexandria : A Study in the History of Greek Algebra. Cambridge, England: Cambridge University Press. p. 286. See especially footnote 4.
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- 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.
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- 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. 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 adopted the Gregorian calendar in 1751, that date became January 30, 1658.]
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- Pell equation solver (n has no upper limit)
- Pell equation solver (n<10^10, can also return the solution to x^2-ny^2 = +-1, +-2, +-3, and +-4)