Type of Diophantine equation
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 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
in his
Br?hmasphu?asiddh?nta
circa 628.
[2]
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. 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
[
edit
]
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
.
[5]
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.
[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
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
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
- and
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's equations with this method, proving that it gives solutions starting from an integer solution of
for
k
= ±1, ±2, or ±4.
[11]
The first general method for solving the Pell's equation (for all
N
) was given by
Bh?skara 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. Bhaskara used it to give the solution
x
=
1
766
319
049
,
y
=
226
153
980
to the
N
= 61 case.
[11]
Several European mathematicians rediscovered how to solve Pell's equation in the 17th century.
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 Frenicle 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
was developed by Lagrange in 1766?1769.
[15]
In particular, Lagrange gave a proof that the Brouncker-Wallis algorithm always terminates.
Solutions
[
edit
]
Fundamental solution via continued fractions
[
edit
]
Let
denote the sequence of
convergents
to the
regular continued fraction
for
. This sequence is unique. Then the pair of positive integers
solving Pell's equation and minimizing
x
satisfies
x
1
=
h
i
and
y
1
=
k
i
for some
i
. This pair is called the
fundamental solution
. The sequence of integers
in the regular continued fraction of
is always eventually periodic. It can be written in the form
, where
is the periodic part repeating indefinitely. Moreover, the tuple
is
palindromic
. It reads the same from left to right as from right to left.
[16]
The fundamental solution is then
The time for finding the fundamental solution using the continued fraction method, with the aid of the
Schonhage?Strassen algorithm
for fast integer multiplication, is within a logarithmic factor of the solution size, the number of digits in the pair
. 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
[
edit
]
Once the fundamental solution is found, all remaining solutions may be calculated algebraically from
[17]
expanding the right side,
equating coefficients
of
on both sides, and equating the other terms on both sides. This yields the
recurrence relations
Concise representation and faster algorithms
[
edit
]
Although writing out the fundamental solution (
x
1
,
y
1
) 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
a
i
,
b
i
, and
c
i
.
For instance,
Archimedes' cattle problem
is equivalent to the Pell equation
, the fundamental solution of which has
206
545
digits if written out explicitly. However, the solution is also equal to
where
and
and
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
where
N
= log
n
is the input size, similarly to the quadratic sieve.
[17]
Quantum algorithms
[
edit
]
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 Vollmer.
[19]
Example
[
edit
]
As an example, consider the instance of Pell's equation for
n
= 7; that is,
The continued fraction of
has the form
. Since the period has length
, which is an even number, the convergent producing the fundamental solution is obtained truncating the continued fraction right before the end of the first occurrence of the period:
.
The sequence of convergents for the square root of seven are
h
/
k
(convergent)
|
h
2
? 7
k
2
(Pell-type approximation)
|
2/1
|
?3
|
3/1
|
+2
|
5/2
|
?3
|
8/3
|
+1
|
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
)
For the Pell's equation
the continued fraction
has a period of odd length. For this the fundamental solution is obtained truncating the continued fraction right before the second occurrence of the period
. Thus, the fundamental solution is
.
The smallest solution can be very large. For example, the smallest solution to
is (
32
188
120
829
134
849
,
1
819
380
158
564
160
), and this is the equation which Frenicle challenged Wallis to solve.
[20]
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
).
(For these records, see
OEIS
:
A033315
for
x
and
OEIS
:
A033319
for
y
.)
List of fundamental solutions of Pell's equations
[
edit
]
The following is a list of the fundamental solution to
with
n
≤ 128. When
n
is an integer square, there is no solution except for the trivial solution (1, 0). The values of
x
are sequence
A002350
and those of
y
are sequence
A002349
in
OEIS
.
n
|
x
|
y
|
1
|
?
|
?
|
2
|
3
|
2
|
3
|
2
|
1
|
4
|
?
|
?
|
5
|
9
|
4
|
6
|
5
|
2
|
7
|
8
|
3
|
8
|
3
|
1
|
9
|
?
|
?
|
10
|
19
|
6
|
11
|
10
|
3
|
12
|
7
|
2
|
13
|
649
|
180
|
14
|
15
|
4
|
15
|
4
|
1
|
16
|
?
|
?
|
17
|
33
|
8
|
18
|
17
|
4
|
19
|
170
|
39
|
20
|
9
|
2
|
21
|
55
|
12
|
22
|
197
|
42
|
23
|
24
|
5
|
24
|
5
|
1
|
25
|
?
|
?
|
26
|
51
|
10
|
27
|
26
|
5
|
28
|
127
|
24
|
29
|
9801
|
1820
|
30
|
11
|
2
|
31
|
1520
|
273
|
32
|
17
|
3
|
n
|
x
|
y
|
33
|
23
|
4
|
34
|
35
|
6
|
35
|
6
|
1
|
36
|
?
|
?
|
37
|
73
|
12
|
38
|
37
|
6
|
39
|
25
|
4
|
40
|
19
|
3
|
41
|
2049
|
320
|
42
|
13
|
2
|
43
|
3482
|
531
|
44
|
199
|
30
|
45
|
161
|
24
|
46
|
24335
|
3588
|
47
|
48
|
7
|
48
|
7
|
1
|
49
|
?
|
?
|
50
|
99
|
14
|
51
|
50
|
7
|
52
|
649
|
90
|
53
|
66249
|
9100
|
54
|
485
|
66
|
55
|
89
|
12
|
56
|
15
|
2
|
57
|
151
|
20
|
58
|
19603
|
2574
|
59
|
530
|
69
|
60
|
31
|
4
|
61
|
1766319049
|
226153980
|
62
|
63
|
8
|
63
|
8
|
1
|
64
|
?
|
?
|
n
|
x
|
y
|
65
|
129
|
16
|
66
|
65
|
8
|
67
|
48842
|
5967
|
68
|
33
|
4
|
69
|
7775
|
936
|
70
|
251
|
30
|
71
|
3480
|
413
|
72
|
17
|
2
|
73
|
2281249
|
267000
|
74
|
3699
|
430
|
75
|
26
|
3
|
76
|
57799
|
6630
|
77
|
351
|
40
|
78
|
53
|
6
|
79
|
80
|
9
|
80
|
9
|
1
|
81
|
?
|
?
|
82
|
163
|
18
|
83
|
82
|
9
|
84
|
55
|
6
|
85
|
285769
|
30996
|
86
|
10405
|
1122
|
87
|
28
|
3
|
88
|
197
|
21
|
89
|
500001
|
53000
|
90
|
19
|
2
|
91
|
1574
|
165
|
92
|
1151
|
120
|
93
|
12151
|
1260
|
94
|
2143295
|
221064
|
95
|
39
|
4
|
96
|
49
|
5
|
n
|
x
|
y
|
97
|
62809633
|
6377352
|
98
|
99
|
10
|
99
|
10
|
1
|
100
|
?
|
?
|
101
|
201
|
20
|
102
|
101
|
10
|
103
|
227528
|
22419
|
104
|
51
|
5
|
105
|
41
|
4
|
106
|
32080051
|
3115890
|
107
|
962
|
93
|
108
|
1351
|
130
|
109
|
158070671986249
|
15140424455100
|
110
|
21
|
2
|
111
|
295
|
28
|
112
|
127
|
12
|
113
|
1204353
|
113296
|
114
|
1025
|
96
|
115
|
1126
|
105
|
116
|
9801
|
910
|
117
|
649
|
60
|
118
|
306917
|
28254
|
119
|
120
|
11
|
120
|
11
|
1
|
121
|
?
|
?
|
122
|
243
|
22
|
123
|
122
|
11
|
124
|
4620799
|
414960
|
125
|
930249
|
83204
|
126
|
449
|
40
|
127
|
4730624
|
419775
|
128
|
577
|
51
|
Connections
[
edit
]
Pell's equation has connections to several other important subjects in mathematics.
Algebraic number theory
[
edit
]
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
.
[21]
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's 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
[
edit
]
Demeyer mentions a connection between Pell's equation and the
Chebyshev polynomials
:
If
and
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
, with
:
[23]
Thus, these polynomials can be generated by the standard technique for Pell's equations of taking powers of a fundamental solution:
It may further be observed that if
are the solutions to any integer Pell's equation, then
and
.
[24]
Continued fractions
[
edit
]
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
.
[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
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
has determinant (?1)
k
.
Smooth numbers
[
edit
]
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's equation
[
edit
]
The negative Pell's equation is given by
and has also been extensively studied. It can be solved by the same method of continued fractions and has 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 4
k
+ 3.
[note 3]
Thus, for example,
x
2
? 3
ny
2
= ?1 is never solvable, but
x
2
? 5
ny
2
= ?1 may be.
[27]
The first few numbers
n
for which
x
2
?
ny
2
= ?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
).
Let
. The proportion of square-free
n
divisible by
k
primes of the form 4
m
+ 1 for which the negative Pell's equation is solvable is at least
α
.
[28]
When the number of prime divisors is not fixed, the proportion is given by 1 -
α.
[29]
[30]
If the negative Pell's 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
As stated above, if the negative Pell's 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
, the next solution is determined in terms of
whenever there is a match, that is, when
is odd. The resulting recursion relation is (modulo a minus sign, which is immaterial due to the quadratic nature of the equation)
which gives an infinite tower of solutions to the negative Pell's equation.
Generalized Pell's equation
[
edit
]
The equation
is called the
generalized
[31]
[32]
(or
general
[16]
)
Pell's equation
. The equation
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
.
[33]
[34]
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
.
[16]
The solution
is called a
Pell multiple
of the solution
.
There exists a finite set of solutions to
such that every solution is a Pell multiple of a solution from that set. In particular, if
is the fundamental solution to
, then each solution to the equation is a Pell multiple of a solution
with
and
, where
.
[35]
If
x
and
y
are positive integer solutions to the Pell's equation with
, then
is a convergent to the continued fraction of
.
[35]
Solutions to the generalized Pell's equation are used for solving certain
Diophantine equations
and
units
of certain
rings
,
[36]
[37]
and they arise in the study of
SIC-POVMs
in
quantum information theory
.
[38]
The equation
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
even, generating a solution to
.
[16]
Notes
[
edit
]
- ^
In Euler's
Vollstandige Anleitung zur Algebra
(pp. 227ff), 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). Vol. 3. Paris, France: Gauthier-Villars et fils.
Letter 17 appears on pp. 457?480. Letter 19 appears on pp. 490?503.
Wallis' letters showing a solution to the Pell's equation also appear in volume 2 of Wallis'
Opera mathematica
(1693), which includes articles by John Pell:
- 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). Vol. 2. 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.
See also:
- ^
Teutsch
is an obsolete form of
Deutsch
, meaning "German". Free E-book:
Teutsche Algebra
at Google Books.
- ^
This is because the Pell equation implies that ?1 is a
quadratic residue
modulo
n
.
References
[
edit
]
- ^
O'Connor, J. J.; Robertson, E. F. (February 2002).
"Pell's Equation"
.
School of Mathematics and Statistics, University of St Andrews, Scotland
. Retrieved
13 July
2020
.
- ^
Dunham, William.
"Number theory ? Number theory in the East"
.
Encyclopedia Britannica
. Retrieved
4 January
2020
.
- ^
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)
(in Latin).
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 a Goldbach, dated 10 August 1750 in:
Fuss, P. H., ed. (1843).
Correspondance Mathematique et Physique de Quelques Celebres Geometres du XVIIIeme Siecle ...
[
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).
Vollstandige 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 Englander, Namens Pell, eine ganz sinnreiche Methode erfunden, welche wir hier erklaren 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 ...
Vol. 2 (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.
- ^
a
b
Tattersall, James (2000).
Elementary Number Theory in Nine Chapters
(PDF)
. Cambridge. p. 274.
doi
:
10.1017/CBO9780511756344
.
ISBN
9780521850148
.
S2CID
118948378
. Archived from
the original
(PDF)
on 15 February 2020.
- ^
a
b
c
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
,
S2CID
120954547
.
- ^
O'Connor, John J.;
Robertson, Edmund F.
,
"Baudhayana"
,
MacTutor History of Mathematics Archive
,
University of St Andrews
- ^
Vardi, I. (1998). "Archimedes' Cattle Problem".
American Mathematical Monthly
.
105
(4). Mathematical Association of America: 305?319.
CiteSeerX
10.1.1.33.4288
.
doi
:
10.2307/2589706
.
JSTOR
2589706
.
- ^
Fraser, Peter M.
(1972).
Ptolemaic Alexandria
. Oxford University Press.
- ^
Weil, Andre
(1972).
Number Theory, an Approach Through History
. Birkhauser.
- ^
Izadi, Farzali (2015).
"Congruent numbers via the Pell equation and its analogous counterpart"
(PDF)
.
Notes on Number Theory and Discrete Mathematics
.
21
: 70?78.
- ^
a
b
John Stillwell
(2002),
Mathematics and its history
(2nd ed.), Springer, pp. 72?76,
ISBN
978-0-387-95336-6
.
- ^
In February 1657, Pierre de Fermat wrote two letters about Pell's equation. One letter (in French) was addressed to Bernard Frenicle de Bessy, and the other (in Latin) was addressed to Kenelm Digby, whom it reached via Thomas White and then William Brouncker.
- Fermat, Pierre de (1894). Tannery, Paul; Henry, Charles (eds.).
Oeuvres de Fermat
(in French and Latin). Vol. 2nd vol. Paris, France: Gauthier-Villars et fils. pp. 333?335.
The letter to Frenicle appears on pp. 333?334; the letter to Digby, on pp. 334?335.
The letter in Latin to Digby is translated into French in:
- Fermat, Pierre de (1896). Tannery, Paul; Henry, Charles (eds.).
Oeuvres de Fermat
(in French and Latin). Vol. 3rd vol. Paris, France: Gauthier-Villars et fils. pp. 312?313.
Both letters are translated (in part) into English in:
- ^
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 destrompe 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 that Britain finally
discarded in 1752
: most of the rest of Europe would have regarded that date as January 31, 1658. See
Old Style and New Style dates#Transposition of historical event dates and possible date conflicts
.
- ^
Rahn, Johann Heinrich (1668) [1659], Brancker, Thomas; Pell (eds.),
An introduction to algebra
.
- ^
"Solution d'un Probleme d'Arithmetique", in
Joseph Alfred Serret
(ed.),
Œuvres de Lagrange
, vol. 1
, pp. 671?731, 1867.
- ^
a
b
c
d
e
f
Andreescu, Titu; Andrica, Dorin (2015).
Quadratic Diophantine Equations
.
New York
: Springer.
ISBN
978-0-387-35156-8
.
- ^
a
b
c
d
Lenstra, H. W. Jr.
(2002),
"Solving the Pell Equation"
(PDF)
,
Notices of the American Mathematical Society
,
49
(2): 182?192,
MR
1875156
.
- ^
Hallgren, Sean (2007), "Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem",
Journal of the ACM
,
54
(1): 1?19,
doi
:
10.1145/1206035.1206039
,
S2CID
948064
.
- ^
Schmidt, A.; Vollmer, U. (2005),
"Polynomial time quantum algorithm for the computation of the unit group of a number field"
(PDF)
,
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing ? STOC '05
, New York: ACM, Symposium on Theory of Computing, pp. 475?480,
CiteSeerX
10.1.1.420.6344
,
doi
:
10.1145/1060590.1060661
,
ISBN
1581139608
,
S2CID
6654142
.
- ^
"Prime Curios!: 313"
.
- ^
Clark, Pete.
"The Pell Equation"
(PDF)
.
University of Georgia
.
- ^
Conrad, Keith.
"Dirichlet's Unit Theorem"
(PDF)
. Retrieved
14 July
2020
.
- ^
Demeyer, Jeroen (2007),
Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields
(PDF)
, PhD thesis,
Ghent University
, p. 70, archived from
the original
(PDF)
on 2 July 2007
, retrieved
27 February
2009
.
- ^
Barbeau, Edward J. (2003), "3. Quadratic surds",
Pell's Equation
, Problem Books in Mathematics, Springer-Verlag,
ISBN
0-387-95529-1
,
MR
1949691
.
- ^
a
b
Størmer, Carl
(1897). "Quelques theoremes sur l'equation de Pell
et leurs applications".
Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl.
(in French).
I
(2).
- ^
Lehmer, D. H.
(1964).
"On a Problem of Størmer"
.
Illinois Journal of Mathematics
.
8
(1): 57?79.
doi
:
10.1215/ijm/1256067456
.
MR
0158849
.
- ^
Wang, Jiaqi; Cai, Lide (December 2013).
"The solvability of negative Pell equation"
(PDF)
.
Tsinghua College
: 5?6.
- ^
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
.
- ^
"Ancient Equations Offer New Look at Number Groups"
.
Quanta Magazine
. 10 August 2022
. Retrieved
18 August
2022
.
- ^
Koymans, Peter; Pagano, Carlo (31 January 2022). "On Stevenhagen's conjecture".
arXiv
:
2201.13424
[
math.NT
].
- ^
Peker, Bilge (2021).
Current Studies in Basic Sciences, Engineering and Technology 2021
(PDF)
. ISRES Publishing. p. 136
. Retrieved
25 February
2024
.
- ^
Tamang, Bal Bahadur (August 2022).
Solvability of Generalized Pell's Equation and Its Applications in Real Life
(PDF)
(Report). Tribhuvan University
. Retrieved
25 February
2024
.
- ^
Lagrange, Joseph-Louis (1867?1892).
Oeuvres de Lagrange. T. 2 / publiees par les soins de M. J.-A. Serret [et G. Darboux]; [precede d'une notice sur la vie et les ouvrages de J.-L. Lagrange, par M. Delambre]
(in French).
- ^
Matthews, Keith.
"The Diophantine Equation
x
2
?
Dy
2
=
N
,
D
> 0"
(PDF)
.
Archived
(PDF)
from the original on 18 March 2015
. Retrieved
20 July
2020
.
- ^
a
b
Conrad, Keith.
"PELL'S EQUATION, II"
(PDF)
. Retrieved
14 October
2021
.
- ^
Bernstein, Leon (1 October 1975). "Truncated units in infinitely many algebraic number fields of degreen ?4".
Mathematische Annalen
.
213
(3): 275?279.
doi
:
10.1007/BF01350876
.
ISSN
1432-1807
.
S2CID
121165073
.
- ^
Bernstein, Leon (1 March 1974).
"On the Diophantine Equation
x
(
x
+
d
)(
x
+ 2
d
) +
y
(
y
+
d
)(
y
+ 2
d
) =
z
(
z
+
d
)(
z
+ 2
d
)"
.
Canadian Mathematical Bulletin
.
17
(1): 27?34.
doi
:
10.4153/CMB-1974-005-5
.
ISSN
0008-4395
.
S2CID
125002637
.
- ^
Appleby, Marcus; Flammia, Steven; McConnell, Gary; Yard, Jon (August 2017). "SICs and Algebraic Number Theory".
Foundations of Physics
.
47
(8): 1042?1059.
arXiv
:
1701.05200
.
Bibcode
:
2017FoPh...47.1042A
.
doi
:
10.1007/s10701-017-0090-7
.
ISSN
0015-9018
.
S2CID
119334103
.
Further reading
[
edit
]
- Edwards, Harold M.
(1996) [1977].
Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory
.
Graduate Texts in Mathematics
. Vol. 50.
Springer-Verlag
.
ISBN
0-387-90230-9
.
MR
0616635
.
- Pinch, R. G. E. (1988).
"Simultaneous Pellian equations"
.
Math. Proc. Cambridge Philos. Soc
.
103
(1): 35?46.
Bibcode
:
1988MPCPS.103...35P
.
doi
:
10.1017/S0305004100064598
.
S2CID
123098216
.
- Whitford, Edward Everett (1912).
The Pell equation
(PhD Thesis)
. Columbia University.
- 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. (eds.).
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
[
edit
]
- Weisstein, Eric W.
"Pell's equation"
.
MathWorld
.
- O'Connor, John J.;
Robertson, Edmund F.
,
"Pell's equation"
,
MacTutor History of Mathematics Archive
,
University of St Andrews
- 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)