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Pell's Equation

2017
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Formalized Mathematics
*

In this article we formalize several basic theorems that correspond to

doi:10.1515/forma-2017-0019
fatcat:jv4wi26vzjgofde5cgguvrjwgy
*Pell's**equation*. ... "Solutions to*Pell's**Equation*" are listed as item #39 from the "Formalizing 100 Theorems" list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. ... Introduction*Pell's**equation*(alternatively called the Pell-Fermat*equation*) is a type of a diophantine*equation*of the form x 2 −Dy 2 = 1 for a natural number D. ...##
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Polynomial Pell's Equations

1976
*
Proceedings of the American Mathematical Society
*

The polynomial

doi:10.2307/2041581
fatcat:avml3gemvvhp7ajh6ujw5s74xq
*Pell's**equation*is P2 -(x1 + d)Q2 = 1, where d is an integer and the solutions P, Q must be polynomials with integer coefficients. ... We consider the polynomial*Pell's**equation*Received by the editors July 1, 1975. AMS (A/05) subject classifications (1970). Primary 10B05, 12A20. ... Then the polynomial*Pell's**equation*P2 -(x2 + d)Q2 = 1 has no nontrivial solution. Theorem 2 . 2 Let d = 1 or d = ±2. ...##
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Polynomial Pell's equation

2002
*
Proceedings of the American Mathematical Society
*

, a necessary and sufficient condition for the polynomial

doi:10.1090/s0002-9939-02-06934-4
fatcat:ecqyaofywrhkxetnpyrsh7spqi
*Pell's**equation*to have a nontrivial solution in Z[x] is obtained. ... We consider the polynomial*Pell's**equation*(1) X 2 − DY 2 = 1 where solutions X, Y are polynomials with integer coefficients. ... Thus if the answer to the second question is negative, then every solution W of the polynomial*Pell's**equation*(1) is expressed as ±W n 0 or ±W n 0 for some n ≥ 1, where W 0 ∈ T 0 . ...##
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Polynomial Pell's equations

1976
*
Proceedings of the American Mathematical Society
*

The polynomial

doi:10.1090/s0002-9939-1976-0401641-4
fatcat:dqaxu6ueybdf3bxhkmhmzqg3em
*Pell's**equation*is P2 -(x1 + d)Q2 = 1, where d is an integer and the solutions P, Q must be polynomials with integer coefficients. ... We consider the polynomial*Pell's**equation*Received by the editors July 1, 1975. AMS (A/05) subject classifications (1970). Primary 10B05, 12A20. ... Then the polynomial*Pell's**equation*P2 -(x2 + d)Q2 = 1 has no nontrivial solution. Theorem 2 . 2 Let d = 1 or d = ±2. ...##
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Polynomial Pell's equation–II

2004
*
Journal of Number Theory
*

Furthermore, all solutions to the polynomial

doi:10.1016/j.jnt.2003.12.005
fatcat:3d3mstbybbcfpfiq3uwxar5jai
*Pell's**equation*satisfying the above condition are determined. r 2004 Elsevier Inc. All rights reserved. ... The polynomial*Pell's**equation*is X 2 À DY 2 ¼ 1; where D is a polynomial with integer coefficients and the solutions X ; Y must be polynomials with integer coefficients. ... We consider the polynomial*Pell's**equation*X 2 À DY 2 ¼ 1; ð1Þ where X ; Y are polynomials with integer coefficients. ...##
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Pell's equation without irrational numbers
[article]

2010
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arXiv
*
pre-print

We solve

arXiv:0806.2490v2
fatcat:ygvtbpvkebcu5kcm2wouv77my4
*Pell's**equation*in a simple way without continued fractions or irrational numbers, and relate the algorithm to the Stern Brocot tree. ... Solving*Pell's**equation*Here is our strategy for generating an infinite number of solutions to*Pell's**equation*x 2 − Dy 2 = 1. 1. First observe that e = (1, 0) T is a solution. ... Introduction For D a positive square-free integer,*Pell's**equation*is x 2 − Dy 2 = 1. This is perhaps the most important Diophantine*equation*. ...##
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Hyperbolic reflections on pell's equation

1989
*
Journal of Number Theory
*

D In this section we establish the connection between

doi:10.1016/0022-314x(89)90064-4
fatcat:ziqddozk5ncfbar6vkp2tmtepu
*Pell's**equation*and the geometry of the reflectors on H/r,*. ... In Section D, the relationship between these reflections and the Pell*equation*is established. This work grew out of consideration of a conjecture about*Pell's**equation*. We now describe this. ...##
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Some new results on negative polynomial Pell's equation
[article]

2021
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arXiv
*
pre-print

The negative polynomial

arXiv:2101.04906v3
fatcat:iwks7uy5b5dchcysuydt53yi6i
*Pell's**equation*is $P^2(X)-(X^2+d)Q^2(X)= -1$, where $d$ is an integer. ... Despite many results on positive polynomial*Pell's**equation*, there is no notable work on negative polynomial*Pell's**equation*, P 2 (X) − D(X)Q 2 (X) = −1, (1.4) where D(X) be a fixed polynomial and P (X ... INTRODUCTION The classical*Pell's**equation*is x 2 − Dy 2 = 1, (1.1) where D is a square-free positive integer. ...##
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Solutions of polynomial Pell's equation

2010
*
Journal of Number Theory
*

, a necessary and sufficient condition for the solution of the polynomial

doi:10.1016/j.jnt.2009.12.006
fatcat:pnamws3wpffkplhikw3ivlrcfy
*Pell's**equation*has been shown. ... In this paper, for the period of the continued fraction of √ D is 4, we show that the polynomial*Pell's**equation*has no nontrivial solutions X, Y ∈ Z[x]. ... With these evidence, we believe that the polynomial*Pell's**equation*(1) Let D = x 4 + ax 3 + bx 2 + cx + d ∈ Z[x] . ...##
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On Polynomial Solutions of Pell's Equation

2021
*
Journal of Mathematics
*

Polynomial

doi:10.1155/2021/5379284
fatcat:fzvkhvorrbdrbjtlqccwovvqtq
*Pell's**equation*is x 2 − D y 2 = ± 1 , where D is a quadratic polynomial with integer coefficients and the solutions X , Y must be quadratic polynomials with integer coefficients. ... In this paper, some quadratic polynomial solutions are given for the*equation*x 2 − D y 2 = ± 1 which are significant from computational point of view. ... (8) constitutes a polynomial solution to*Pell's**equation*. ...##
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Method of Hidden Parameters and Pell's Equation

2002
*
Zenodo
*

Using the representation of numbers, the methods of decreasing the number of calculation steps for

doi:10.5281/zenodo.3463927
fatcat:ttvywgeeizdf5eynh7csohzomy
*Pell's**equation*are developed. ... The parametric representation of*Pell's**equation*solutions are obtained with the help of new "method of hidden parameters". ...*Pell's**equation*may seem to be fully studied and no new phenomena can be found. ...##
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The Solvability of Polynomial Pell's Equation

2020
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Journal of Institute of Science and Technology
*

As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial

doi:10.3126/jist.v25i2.33749
fatcat:e6dsomtrpjgozgc3rxqt4tshvm
*Pell's**equation*p2-Dq2=1 where D=f2+2g is monic quadratic polynomial with deg g<deg f and the solutions ... So,*Pell's**equation*is studied in number theory. ... RESULTS Solvability of Polynomial*Pell's**equation*We begin by exploring some well-known basic properties of the*Pell's**equation*over polynomials, usually called the polynomial*Pell's**equation*. ...##
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Method of Hidden Parameters and Pell's Equation
[article]

2003
*
arXiv
*
pre-print

Using the representation of numbers, the methods of decreasing the number of calculation steps for

arXiv:math/0303176v1
fatcat:anmhodnflnhifagvs3lo4tvr3m
*Pell's**equation*are developed. ... The parametric representation of*Pell's**equation*solutions are obtained with the help of new "method of hidden parameters". ...*Pell's**equation*may seem to be fully studied and no new phenomena can be found. ...##
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On the least solution of Pell's equation

1942
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Bulletin of the American Mathematical Society
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Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents
[article]

2019
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arXiv
*
pre-print

In this paper we give describe a new connection between the dilogarithm function and solutions to

arXiv:1903.04074v3
fatcat:gj3hjhrwo5hblk7xwbiig7wppi
*Pell's**equation*$x^2-ny^2 = \pm 1$. ... For each solution $x,y$ to*Pell's**equation*we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit $x+y\sqrt{n} \in \Z[\sqrt{n}]$. ... over all solutions to*Pell's**equation*. ...
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