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Pell's Equation

Marcin Acewicz, Karol Pąk
2017 Formalized Mathematics  
In this article we formalize several basic theorems that correspond to 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.  ... 
doi:10.1515/forma-2017-0019 fatcat:jv4wi26vzjgofde5cgguvrjwgy

Polynomial Pell's Equations

Melvyn B. Nathanson
1976 Proceedings of the American Mathematical Society  
The polynomial 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.  ... 
doi:10.2307/2041581 fatcat:avml3gemvvhp7ajh6ujw5s74xq

Polynomial Pell's equation

William A. Webb, Hisashi Yokota
2002 Proceedings of the American Mathematical Society  
, a necessary and sufficient condition for the polynomial 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 .  ... 
doi:10.1090/s0002-9939-02-06934-4 fatcat:ecqyaofywrhkxetnpyrsh7spqi

Polynomial Pell's equations

Melvyn B. Nathanson
1976 Proceedings of the American Mathematical Society  
The polynomial 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.  ... 
doi:10.1090/s0002-9939-1976-0401641-4 fatcat:dqaxu6ueybdf3bxhkmhmzqg3em

Polynomial Pell's equation–II

W.A. Webb, H. Yokota
2004 Journal of Number Theory  
Furthermore, all solutions to the polynomial 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.  ... 
doi:10.1016/j.jnt.2003.12.005 fatcat:3d3mstbybbcfpfiq3uwxar5jai

Pell's equation without irrational numbers [article]

N. J. Wildberger
2010 arXiv   pre-print
We solve 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.  ... 
arXiv:0806.2490v2 fatcat:ygvtbpvkebcu5kcm2wouv77my4

Hyperbolic reflections on pell's equation

Mark Sheingorn
1989 Journal of Number Theory  
D In this section we establish the connection between 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.  ... 
doi:10.1016/0022-314x(89)90064-4 fatcat:ziqddozk5ncfbar6vkp2tmtepu

Some new results on negative polynomial Pell's equation [article]

K. Anitha, I. Mumtaj Fathima, A R Vijayalakshmi
2021 arXiv   pre-print
The negative polynomial 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.  ... 
arXiv:2101.04906v3 fatcat:iwks7uy5b5dchcysuydt53yi6i

Solutions of polynomial Pell's equation

H. Yokota
2010 Journal of Number Theory  
, a necessary and sufficient condition for the solution of the polynomial 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] .  ... 
doi:10.1016/j.jnt.2009.12.006 fatcat:pnamws3wpffkplhikw3ivlrcfy

On Polynomial Solutions of Pell's Equation

Hasan Sankari, Ahmad Abdo, Marco Fontana
2021 Journal of Mathematics  
Polynomial 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.  ... 
doi:10.1155/2021/5379284 fatcat:fzvkhvorrbdrbjtlqccwovvqtq

Method of Hidden Parameters and Pell's Equation

S.N. Arteha
2002 Zenodo  
Using the representation of numbers, the methods of decreasing the number of calculation steps for 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.  ... 
doi:10.5281/zenodo.3463927 fatcat:ttvywgeeizdf5eynh7csohzomy

The Solvability of Polynomial Pell's Equation

Bal Bahadur Tamang, Ajay Singh
2020 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 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.  ... 
doi:10.3126/jist.v25i2.33749 fatcat:e6dsomtrpjgozgc3rxqt4tshvm

Method of Hidden Parameters and Pell's Equation [article]

S.N. Arteha
2003 arXiv   pre-print
Using the representation of numbers, the methods of decreasing the number of calculation steps for 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.  ... 
arXiv:math/0303176v1 fatcat:anmhodnflnhifagvs3lo4tvr3m

On the least solution of Pell's equation

Loo-keng Hua
1942 Bulletin of the American Mathematical Society  
doi:10.1090/s0002-9904-1942-07768-8 fatcat:kvbb77yuh5aebkueqa7x2rek3a

Dilogarithm identities for solutions to Pell's equation in terms of continued fraction convergents [article]

Martin Bridgeman
2019 arXiv   pre-print
In this paper we give describe a new connection between the dilogarithm function and solutions to 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.  ... 
arXiv:1903.04074v3 fatcat:gj3hjhrwo5hblk7xwbiig7wppi
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