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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 is represented by 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,^[1] who found an integer solution to $92x^{2}+1=y^{2}$ in his Brāhmasphuṭasiddhānta circa 628.^[2] 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. William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose from Leonhard Euler mistakenly attributing Brouncker's solution of the equation to John Pell.^[3]^[4]^{[note 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 2.^[5] Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of $\sqrt 2$ . 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.^[5] 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.^[6]

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.^[5] 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,^[7] and the attribution to Archimedes is generally accepted today.^[8]^[9]

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.^[10]

In Indian mathematics, Brahmagupta discovered that

(x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2},

a form of what is now known as 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 triples

(x_{1}x_{2}+Ny_{1}y_{2},x_{1}y_{2}+x_{2}y_{1},k_{1}k_{2})

and

(x_{1}x_{2}-Ny_{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 $x^{2}-Ny^{2}=1$ 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, proving that it gives solutions starting from an integer solution of $x^{2}-Ny^{2}=k$ for k = ±1, ±2, or ±4.^[11]

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 $a$ 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 $(am+Nb,a+bm,k(m^{2}-N))$ , 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 ${\frac {a+bm}{k}}$ is an integer, so are the other two numbers in the triple. Among such $m$ , the method chooses one that minimizes ${\frac {m^{2}-N}{k}}$ , 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.^[11]

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.^[12] 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.^[13]

John Pell's connection with the equation is that he revised Thomas Branker's translation^[14] 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.^[4]

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.^[15]

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 $(x_{1},y_{1})$ 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.^[16]

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 $(x_{1},y_{1})$ . 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.^[17]

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},

^[17]

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}+ny_{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

u=x'_{1}+y'_{1}{\sqrt {4729494}}=(300426607914281713365{\sqrt {609}}+84129507677858393258{\sqrt {7766}})^{2}

and $x'_{1}$ and $y'_{1}$ only have 45 and 41 decimal digits, respectively.^[17]

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.^[17]

Quantum algorithms

Hallgren showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time.^[18] 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.^[19]

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)	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 $x^{2}-313y^{2}=1$ is (32188120829134849, 1819380158564160), and this is the equation which Frenicle challenged Wallis to solve.^[20] 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 OEIS: A033315 for x and OEIS: A033319 for y.)

List of fundamental solutions of Pell 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 A002350 and those of y are sequence A002349 in OEIS, or A033313 and A033317 in OEIS with square n skipped.

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}-ny^{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 $\mathbb {Z} [{\sqrt {n}}]$ .^[21] 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.^[22] 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: 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 $n=x^{2}-1$ :^[23]

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 $x_{i}=T_{i}(x_{1})$ and $y_{i}=y_{1}U_{i-1}(x_{1})$ .^[24]

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.^[16]

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.

Smooth numbers

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.^[25]^[26] 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.^[25]

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 if and only if the length of the period of the continued fraction of ${\sqrt {n}}$ (OEIS: A003285) is odd. 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 and x² − 4ny² = −1 are never solvable, but x² − 5ny² = −1 may be,^[27] such as when n = 13 or 17, though not when n = 41 or 61.

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).

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%.^[28] 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}-ny^{2})^{2}=(-1)^{2}

implies

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

As stated above, if the negative Pell 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 $(x+{\sqrt {n}}y)(x-{\sqrt {n}}y)=-1$ , the next solution is determined in terms of $\imath (x_{k}+{\sqrt {n}}y_{k})=(\imath (x+{\sqrt {n}}y))^{k}$ 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_{k-2}x_{1}^{2}+nx_{k-2}y_{1}^{2}+2ny_{k-2}y_{1}x_{1}

y_{k}=y_{k-2}x_{1}^{2}+ny_{k-2}y_{1}^{2}+2x_{k-2}y_{1}x_{1}

,

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

The following is a list of the smallest solution to $x^{2}-ny^{2}=-1$ with n ≤ 512, if solvable. The values of x are sequence A130226 and those of y are sequence A130227 in OEIS, these two OEIS sequence only list the n such that this equation is solvable.

n	x	y	n	x	y	n	x	y	n	x	y
2	1	1	106	4005	389	241	71011068	4574225	365	3458	181
5	2	1	109	8890182	851525	250	4443	281	370	327	17
10	3	1	113	776	73	257	16	1	373	5118	265
13	18	5	122	11	1	265	6072	373	389	1282	65
17	4	1	125	682	61	269	82	5	394	395023035	19900973
26	5	1	130	57	5	274	1407	85	397	20478302982	1027776565
29	70	13	137	1744	149	277	8920484118	535979945	401	20	1
37	6	1	145	12	1	281	1063532	63445	409	111921796968	5534176685
41	32	5	149	113582	9305	290	17	1	421	44042445696821418	2146497463530785
50	7	1	157	4832118	385645	293	2482	145	425	268	13
53	182	25	170	13	1	298	409557	23725	433	7230660684	347483377
58	99	13	173	1118	85	313	126862368	7170685	442	21	1
61	29718	3805	181	1111225770	82596761	314	443	25	445	4662	221
65	8	1	185	68	5	317	352618	19805	449	189471332	8941705
73	1068	125	193	1764132	126985	325	18	1	457	59089951584	2764111349
74	43	5	197	14	1	337	1015827336	55335641	458	107	5
82	9	1	202	3141	221	338	239	13	461	24314110	1132421
85	378	41	218	251	17	346	93	5	481	964140	43961
89	500	53	226	15	1	349	9210	493	485	22	1
97	5604	569	229	1710	113	353	71264	3793	493	683982	30805
101	10	1	233	23156	1517	362	19	1	509	395727950	17540333

For the smallest solutions of x²−ny² = ±1, the values of x are sequence A006702 and those of y are sequence A006703 in OEIS, or A077232 and A077233 in OEIS with square n skipped.

Generalized Pell's equation

The equation

x^{2}-dy^{2}=N

is called the generalized^{[citation needed]} (or general^[16]) Pell's equation. Like $x^{2}-dy^{2}=-1$ , cases such as $x^{2}-dy^{2}=2$ and $x^{2}-dy^{2}=-2$ are not always solvable (see OEIS sequence A261246 and this generalized Pell equation solver). The equation $u^{2}-dv^{2}=1$ is the corresponding Pell's resolvent.^[16] A recursive algorithm was given by Lagrange in 1768 for solving the equation, reducing the problem to the case $\left\vert N\right\vert <{\sqrt {d}}$ .^[29]^[30] 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 $x_{n}+y_{n}{\sqrt {d}}=(x_{0}+y_{0}{\sqrt {d}})(u_{n}+v_{n}{\sqrt {d}})$ is a solution to $x^{2}-dy^{2}=N$ , a principle named the multiplicative principle.^[16]

Solutions to the generalized Pell's equation are used for solving certain Diophantine equations and units of certain rings,^[31]^[32] and they arise in the study of SIC-POVMs in quantum information theory.^[33]