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Pell's equation (also called the Pell–Fermat equation) is any Diophantine equation of the form
Contents
- History
- Fundamental solution via continued fractions
- Additional solutions from the fundamental solution
- Concise representation and faster algorithms
- Quantum algorithms
- Example
- The smallest solution of Pell equations
- Connections
- Algebraic number theory
- Chebyshev polynomials
- Continued fractions
- The negative Pell equation
- Transformations
- References
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 developed the chakravala method to solve Pell's equation and other quadratic indeterminate equations in his 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 the Pell 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 to a similar date in India. The name of Pell's equation arose from Leonhard Euler's mistakenly attributing Lord Brouncker's solution of the equation to John Pell.
History
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 two. Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √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 two.
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. Archimedes' cattle problem involves solving a Pellian equation. It is now generally accepted that this problem is due to Archimides.
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
(see Brahmagupta's identity). Using this, he was able to "compose" triples
Not only did this give a way to generate infinitely many solutions to
The first general method for solving the Pell equation (for all N) was given by Bhaskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method, it starts by composing any triple
When
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. Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to 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 Lord Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.
Pell's connection with the equation is that he revised Thomas Branker's translation (Rahn 1668) of Johann Rahn's 1659 book "Teutsche Algebra" into English, with a discussion of Brouncker's solution of the equation. 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 fractions and algebraic manipulations with numbers of the form
Fundamental solution via continued fractions
Let
As 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 (x1,y1). However, this is not a polynomial time algorithm because the number of digits in the solution may be as large as √n, far larger than a polynomial in the number of digits in the input value n (Lenstra 2002).
Additional solutions from the fundamental solution
Once the fundamental solution is found, all remaining solutions may be calculated algebraically from
expanding the right side, equating coefficients of
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
where
and
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 √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
where N = log n is the input size, similarly to the quadratic sieve (Lenstra 2002).
Quantum algorithms
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 Schmidt & Völlmer (2005).
Example
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
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
(For these records, see A033315 (x), and A033319 (y)).
The smallest solution of Pell equations
The following is a list of the smallest solution to
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 numbers, as the formula
is the norm for the ring
Chebyshev polynomials
Demeyer (2007) 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) (Barbeau, chapter 3).
Continued fractions
A general development of solutions of Pell's equation
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 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. 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.
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
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. 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, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, ... (sequence A031396 in the OEIS).Cremona & Odoni (1989) demonstrate that 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:
implies
Transformations
The related equation
can be used to find solutions to the positive Pell equation for certain d. Legendre proved that all primes of form d = 4m + 3 solve one case of (*), with the form 8m + 3 solving the negative, and 8m + 7 for the positive. Their fundamental solution then leads to the one for x2−dy2 = 1. This can be shown by squaring both sides of (*)
to get
Since
and so the fundamental solutions to (*) are smaller than those of the associated negative Pell equation. For example, u2-3v2 = -2 is {u,v} = {1, 1}, so x2 − 3y2 = 1 has {x,y} = {2, 1}. On the other hand, u2 − 7v2 = 2 is {u,v} = {3,1}, so x2 − 7y2 = 1 has {x,y} = {8,3}.
Another related equation,
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, if fundamental {u,v} are both odd, then it leads to fundamental {x,y}.
1. If u2 − dv2 = −4, and {x,y} = {(u2 + 3)u/2, (u2 + 1)v/2}, then x2 − dy2 = −1.
Ex. Let d = 13, then {u,v} = {3, 1} and {x,y} = {18, 5}.
2. If u2 − dv2 = 4, and {x,y} = {(u2 − 3)u/2, (u2 − 1)v/2}, then x2 − dy2 = 1.
Ex. Let d = 13, then {u,v} = {11, 3} and {x,y} = {649, 180}.
3. If u2 − dv2 = −4, and {x,y} = {(u4 + 4u2 + 1)(u2 + 2)/2, (u2 + 3)(u2 + 1)uv/2}, then x2 − dy2 = 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 u.