Pell's equation

Pell's equation for n = 2 and six of its integer solutions

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 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.^[1]

History

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

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

and from the closely related equation

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

because of the connection of these equations to the square root of two.^[2] 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.^[2] 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.^[2] Archimedes' cattle problem involves solving a Pellian equation. It is now generally accepted that this problem is due to Archimides. ^[3][4]

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

$(x_1^2 - Ny_1^2)(x_2^2 - Ny_2^2) = (x_1x_2 + Ny_1y_2)^2 - N(x_1y_2 + x_2y_1)^2 = (x_1x_2 - Ny_1y_2)^2 - N(x_1y_2 - x_2y_1)^2$

(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 $x^{2}-Ny^{2}=k$, to generate the new triple

$(x_1x_2 + Ny_1y_2 \,,\, x_1y_2 + x_2y_1 \,,\, k_1k_2)$ 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 $y$, being squared) 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 $x^{2}-Ny^{2}=k$ for $k$ = ±1, ±2, or ±4.^[5]

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 $a^2 - Nb^2 = k$) with the trivial triple $(m, 1, m^2 - N)$ to get the triple $(am + Nb, a+bm, k(m^2-N))$, which can be scaled down to

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

When $m$ is chosen so that ${\displaystyle (a+bm)/k}$ is an integer, so are the other two numbers in the triple. Among such $m$, the method chooses one that minimizes ${\displaystyle (m^{2}-N)/k}$, 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.^[5]

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 $P+Q\sqrt{a},$ was developed by Lagrange in 1766–1769.^[6]

Solutions

Fundamental solution via continued fractions

Let $\tfrac{h_i}{k_i}$ denote the sequence of convergents to the regular continued fraction for ${\sqrt {n}}$. 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

${\displaystyle x_{k}+y_{k}{\sqrt {n}}=(x_{1}+y_{1}{\sqrt {n}})^{k},}$

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

$\displaystyle x_{k+1} = x_1 x_k + n y_1 y_k,$

$\displaystyle y_{k+1} = x_1 y_k + y_1 x_k.$

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

$x_1+y_1\sqrt n = \prod_{i=1}^t (a_i + b_i\sqrt n)^{c_i}$

using much smaller integers a_i, b_i, and c_i.

For instance, Archimedes' cattle problem is equivalent to the Pell equation $x^{2}-410286423278424y^{2}=1$, the fundamental solution of which has 206545 digits if written out explicitly. However, the solution is also equal to

$x_1+y_1\sqrt n=u^{2329},$

where

${\displaystyle u=x'_{1}+y'_{1}{\sqrt {4729494}}}$

and ${\displaystyle x'_{1}}$ and ${\displaystyle y'_{1}}$ only have 45 and 41 decimal digits, respectively. (Alternatively, one may write even more concisely

$u = (300426607914281713365\sqrt{609}+84129507677858393258\sqrt{7766})^2.$ ^[7])

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

$\exp O(\sqrt{\log N\log\log N}),$

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,

$x^{2}-7y^{2}=1$

The sequence of convergents for the square root of seven are

h / k (Convergent)	h^2 −7k^2 (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 $x^2-313y^2 = 1$ is (32188120829134849, 1819380158564160), and this is the equation which Frenicle challenged Wallis to solve.^[8] 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, ... (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 $x^2 - ny^2 = 1$ with n ≤ 128. For square n, there are no solutions except (1, 0). (sequence A002350 (x) and A002349 (y) in OEIS, or  A033313 (x) and  A033317 (y) (for nonsquare n))

n	x	y	n	x	y	n	x	y	n	x	y
1	-	-	33	23	4	65	129	16	97	62809633	6377352
2	3	2	34	35	6	66	65	8	98	99	10
3	2	1	35	6	1	67	48842	5967	99	10	1
4	-	-	36	-	-	68	33	4	100	-	-
5	9	4	37	73	12	69	7775	936	101	201	20
6	5	2	38	37	6	70	251	30	102	101	10
7	8	3	39	25	4	71	3480	413	103	227528	22419
8	3	1	40	19	3	72	17	2	104	51	5
9	-	-	41	2049	320	73	2281249	267000	105	41	4
10	19	6	42	13	2	74	3699	430	106	32080051	3115890
11	10	3	43	3482	531	75	26	3	107	962	93
12	7	2	44	199	30	76	57799	6630	108	1351	130
13	649	180	45	161	24	77	351	40	109	158070671986249	15140424455100
14	15	4	46	24335	3588	78	53	6	110	21	2
15	4	1	47	48	7	79	80	9	111	295	28
16	-	-	48	7	1	80	9	1	112	127	12
17	33	8	49	-	-	81	-	-	113	1204353	113296
18	17	4	50	99	14	82	163	18	114	1025	96
19	170	39	51	50	7	83	82	9	115	1126	105
20	9	2	52	649	90	84	55	6	116	9801	910
21	55	12	53	66249	9100	85	285769	30996	117	649	60
22	197	42	54	485	66	86	10405	1122	118	306917	28254
23	24	5	55	89	12	87	28	3	119	120	11
24	5	1	56	15	2	88	197	21	120	11	1
25	-	-	57	151	20	89	500001	53000	121	-	-
26	51	10	58	19603	2574	90	19	2	122	243	22
27	26	5	59	530	69	91	1574	165	123	122	11
28	127	24	60	31	4	92	1151	120	124	4620799	414960
29	9801	1820	61	1766319049	226153980	93	12151	1260	125	930249	83204
30	11	2	62	63	8	94	2143295	221064	126	449	40
31	1520	273	63	8	1	95	39	4	127	4730624	419775
32	17	3	64	-	-	96	49	5	128	577	51

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

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

is the norm for the ring $\mathbb{Z}[\sqrt{n}]$ and for the closely related quadratic field $\mathbb{Q}(\sqrt{n})$. Thus, a pair of integers $(x,y)$ solves Pell's equation if and only if $x + y \sqrt{n}$ is a unit with norm 1 in $\mathbb{Z}[\sqrt{n}]$. Dirichlet's unit theorem, that all units of $\mathbb{Z}[\sqrt{n}]$ 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 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:

$T_i + U_{i-1} \sqrt{x^2-1} = (x + \sqrt{x^2-1})^i. \,$

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 $x^{2}-ny^{2}=1$ in terms of continued fractions of ${\sqrt {n}}$ 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

$\begin{pmatrix} p & q \\ nq & p \end{pmatrix}$

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

$\begin{pmatrix} p_{k-1} & p_{k} \\ q_{k-1} & q_{k} \end{pmatrix}$

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

$x^{2}-ny^{2}=-1$

It has also been extensively studied; it can be solved by the same method of continued fractions and will have solutions when 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 4m + 3.^[9] Thus, for example, x² − 3py² = −1 is never solvable, but x² − 5py² = −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:

${\displaystyle (x^{2}-ny^{2})^{2}=(-1)^{2}}$

implies

${\displaystyle (x^{2}+ny^{2})^{2}-n(2xy)^{2}=1.}$

Transformations

The related equation

${\displaystyle u^{2}-dv^{2}=\pm 2}$ (*)

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 (*)

${\displaystyle (u^{2}-dv^{2})^{2}=(\pm 2)^{2}}$

to get

${\displaystyle (u^{2}+dv^{2})^{2}-d(2uv)^{2}=4.}$

Since $dv^2 = u^2 \mp 2$ from (*), it follows that

${\displaystyle (u^{2}\mp 1)^{2}-d(uv)^{2}=1,}$

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,

${\displaystyle u^{2}-dv^{2}=\pm 4}$

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,^[10] 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.

Notes

1. Lettre IX. Euler à Goldbach, dated 10 August 1750 in: P. H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 35-39 ; see especially page 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.)
2. Knorr, Wilbur R. (1976), "Archimedes and the measurement of the circle: a new interpretation", Archive for History of Exact Sciences, 15 (2): 115–140, doi:10.1007/bf00348496, MR 0497462.
3. Fraser, P.M. (1972). Ptolemaic Alexandria. Oxford University Press.
4. Weil, A. (1972). Number Theory, an Approach Through History. Birkhäuser.
5. John Stillwell (2002), Mathematics and its history (2nd ed.), Springer, pp. 72–76, ISBN 978-0-387-95336-6
6. "Solution d'un Problème d'Arithmétique", in J.-A. Serret (Ed.), Oeuvres de Lagrange, vol. 1, pp. 671–731, 1867.
7. (Lenstra 2002)
8. Prime Curios!: 313
9. This is because the Pell equation implies that −1 is a quadratic residue modulo n.
10. A Collection of Algebraic Identities: Pell Equations.

References

• Barbeau, Edward J. (2003), Pell's Equation, Problem Books in Mathematics, Springer-Verlag, ISBN 0-387-95529-1, MR 1949691.
• Whitford, Edward Everett (1912), The Pell equation (Phd Thesis), Columbia University
• Cremona, John E.; Odoni, R. W. K. (1989), "Some density results for negative Pell equations; an application of graph theory", Journal of the London Mathematical Society. Second Series, 39 (1): 16–28, doi:10.1112/jlms/s2-39.1.16, ISSN 0024-6107.
• Demeyer, Jeroen (2007), Diophantine Sets over Polynomial Rings and Hilbert’s Tenth Problem for Function Fields (PDF), Ph.D. thesis, Universiteit Gent, p. 70.
• Edwards, Harold M. (1996), Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, Graduate Texts in Mathematics, 50, Springer-Verlag, ISBN 0-387-90230-9, MR 0616635. Originally published 1977.
• Hallgren, Sean (2007), "Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem", Journal of the ACM, 54 (1): Art. No. 4, doi:10.1145/1206035.1206039..
• Lenstra, H. W., Jr. (2002), "Solving the Pell Equation" (PDF), Notices of the American Mathematical Society, 49 (2): 182–192, MR 1875156..
• Pinch, R. G. E. (1988), "Simultaneous Pellian equations", Math. Proc. Cambridge Philos. Soc., 103 (1): 35–46, doi:10.1017/S0305004100064598.
• Rahn, Johann Heinrich (1668) [1659], Brancker, Thomas; Pell, eds., An introduction to algebra
• Schmidt, A.; Völlmer, U. (2005), "Polynomial time quantum algorithm for the computation of the unit group of a number field", Proc. 37th Annual ACM Symposium on Theory of Computing, New York: ACM, pp. 475–480, doi:10.1145/1060590.1060661.
• Wildberger, N.J., Divine Proportions : Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney, 2005.

Further reading

• Williams, H. C. (2002). "Solving the Pell equation". In Bennett, M. A.; Berndt, B.C.; Boston, N.; Diamond, H.G.; Hildebrand, A.J.; Philipp, W. Surveys in number theory: Papers from the millennial conference on number theory. Natick, MA: A K Peters. pp. 325–363. ISBN 1-56881-162-4. Zbl 1043.11027.

External links

• Weisstein, Eric W. "Pell's equation". MathWorld.

• Pell's equation
• 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)