Pell's equation

History

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 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

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, Brahmagupta discovered that

( x 1 2 − N y 1 2 ) ( x 2 2 − N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 − N ( x 1 y 2 + x 2 y 1 ) 2 = ( x 1 x 2 − N y 1 y 2 ) 2 − N ( x 1 y 2 − x 2 y 1 ) 2

(see 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 − N y 2 = k , to generate the new triples

( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) and ( x 1 x 2 − N y 1 y 2 , x 1 y 2 − x 2 y 1 , k 1 k 2 ) .

Not only did this give a way to generate infinitely many solutions to x 2 − N y 2 = 1 starting with one solution, but also, by dividing such a composition by k 1 k 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 equations with this method; in particular he showed how to obtain solutions starting from an integer solution of x 2 − N y 2 = k for k = ±1, ±2, or ±4.

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 ( a , b , k ) (that is, one which satisfies a 2 − N b 2 = k ) with the trivial triple ( m , 1 , m 2 − N ) to get the triple ( a m + N b , a + b m , k ( m 2 − N ) ) , which can be scaled down to

( a m + N b k , a + b m k , m 2 − N k ) .

When m is chosen so that ( a + b m ) / k is an integer, so are the other two numbers in the triple. Among such m , the method chooses one that minimizes ( m 2 − N ) / k , and repeats the process. This method always terminates with a solution (proved by Lagrange in 1768). Bhaskara used it to give the solution x = 1766319049, y = 226153980 to the notorious 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. 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 P + Q a , was developed by Lagrange in 1766–1769.

Fundamental solution via continued fractions

Let h i k i denote the sequence of convergents to the regular continued fraction for n . This sequence is unique. Then the pair (x₁,y₁) 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. Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found.

As 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

x k + y k n = ( x 1 + y 1 n ) k ,

expanding the right side, equating coefficients of 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 .

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

x 1 + y 1 n = ∏ i = 1 t ( a i + b i n ) 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 − 410286423278424 y 2 = 1 , the fundamental solution of which has 206545 digits if written out explicitly. However, the solution is also equal to

x 1 + y 1 n = u 2329 ,

where

u = x 1 ′ + y 1 ′ 4729494

and x 1 ′ and y 1 ′ only have 45 and 41 decimal digits, respectively. (Alternatively, one may write even more concisely

u = ( 300426607914281713365 609 + 84129507677858393258 7766 ) 2 . )

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

exp ⁡ O ( log ⁡ N log ⁡ log ⁡ N ) ,

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,

x 2 − 7 y 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 A001081 (x) and A001080 (y) in OEIS)

The smallest solution can be very large. For example, the smallest solution to x 2 − 313 y 2 = 1 is (32188120829134849, 1819380158564160), and this is the equation which Frenicle challenged Wallis to solve. Values of n such that the smallest solution of x 2 − n y 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, ... (sequence A033316 in the OEIS)

(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 x 2 − n y 2 = 1 with n ≤ 128. For square n, there is no solution except (1, 0). (sequence A002350 (x) and A002349 (y) in OEIS, or  A033313 (x) and  A033317 (y) (for nonsquare n))

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

x 2 − n y 2 = ( x + y n ) ( x − y n )

is the norm for the ring Z [ n ] and for the closely related quadratic field Q ( n ) . Thus, a pair of integers ( x , y ) solves Pell's equation if and only if x + y n is a unit with norm 1 in Z [ n ] . Dirichlet's unit theorem, that all units of Z [ 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 equation can be generated from the fundamental solution. The fundamental unit can in general be found by solving a Pell-like equation but it does not always correspond directly to the fundamental solution of Pell's equation itself, because the fundamental unit may have norm −1 rather than 1 and its coefficients may be half integers rather than integers.

Chebyshev polynomials

Demeyer (2007) mentions a connection between Pell's equation and the Chebyshev polynomials: If T_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 = x² − 1:

T i 2 − ( x 2 − 1 ) U i − 1 2 = 1.

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

T i + U i − 1 x 2 − 1 = ( x + x 2 − 1 ) i .

It may further be observed that, if (x_i,y_i) are the solutions to any integer Pell equation, then x_i = T_i (x₁) and y_i = y₁U_i − 1(x₁) (Barbeau, chapter 3).

Continued fractions

A general development of solutions of Pell's equation x 2 − n y 2 = 1 in terms of continued fractions of 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

( p q n q p )

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

( p k − 1 p k q k − 1 q k )

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

x 2 − n y 2 = − 1

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:

( x 2 − n y 2 ) 2 = ( − 1 ) 2

implies

( x 2 + n y 2 ) 2 − n ( 2 x y ) 2 = 1.

Transformations

The related equation

u 2 − d v 2 = ± 2 (*)

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 (*)

( u 2 − d v 2 ) 2 = ( ± 2 ) 2

to get

( u 2 + d v 2 ) 2 − d ( 2 u v ) 2 = 4.

Since d v 2 = u 2 ∓ 2 from (*), it follows that

( u 2 ∓ 1 ) 2 − d ( u v ) 2 = 1 ,

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,

u 2 − d v 2 = ± 4

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.