The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. 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 p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

scholarly journals ON GENERALIZED THUE–MORSE FUNCTIONS AND THEIR VALUES

2019 ◽  
Vol 108 (2) ◽  
pp. 177-201
Author(s):  
DZMITRY BADZIAHIN ◽  
EVGENIY ZORIN
Keyword(s):  
Generating Function ◽  
Continued Fraction ◽  
Laurent Series ◽  
Regular Structure ◽  
Continued Fraction Expansion ◽  
Infinite Products ◽  
Morse Functions ◽  
Morse Number ◽  
Badly Approximable

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, $d\in \mathbb{N}$, $d\geq 2$, which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\geq 2$, are badly approximable.

scholarly journals On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

2018 ◽  
Vol 61 (1) ◽  
pp. 283-293
Author(s):  
Poj Lertchoosakul ◽  
Radhakrishnan Nair
Keyword(s):  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Laurent Series ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series ◽  
Almost Everywhere ◽  
Partial Quotients ◽  
Image Position

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

scholarly journals A criterion for periodicity of multi-continued fraction expansion of multi-formal Laurent series

2007 ◽  
Vol 130 (2) ◽  
pp. 127-140 ◽  
Author(s):  
Zongduo Dai ◽  
Ping Wang ◽  
Kunpeng Wang ◽  
Xiutao Feng
Keyword(s):  
Continued Fraction ◽  
Laurent Series ◽  
Continued Fraction Expansion ◽  
Formal Laurent Series

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field

2013 ◽  
Vol 56 (2) ◽  
pp. 258-264
Author(s):  
A. Chandoul ◽  
M. Jellali ◽  
M. Mkaouar
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Laurent Series ◽  
Minimal Polynomial ◽  
Ordered Set ◽  
Pisot Number ◽  
Continued Fraction Expansion ◽  
Formal Power

Abstract.Dufresnoy and Pisot characterized the smallest Pisot number of degree n ≥ 3 by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot’s result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree n in the field of formal power series over a finite field is given by P(Y) = Yn–XYn-1–αn where α is the least element of the finite field 픽q\{0} (as a finite total ordered set). We prove that the sequence of SPEs of degree n is decreasing and converges to αX: Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

scholarly journals Comparative Study of Discrete Component Realizations of Fractional-Order Capacitor and Inductor Active Emulators

2018 ◽  
Vol 27 (11) ◽  
pp. 1850170 ◽  
Author(s):  
Georgia Tsirimokou ◽  
Aslihan Kartci ◽  
Jaroslav Koton ◽  
Norbert Herencsar ◽  
Costas Psychalinos
Keyword(s):  
Comparative Study ◽  
Fractional Order ◽  
High Performance ◽  
Continued Fraction ◽  
Transfer Functions ◽  
Operational Amplifiers ◽  
Current Conveyors ◽  
Continued Fraction Expansion ◽  
Current Feedback ◽  
Discrete Component

Due to the absence of commercially available fractional-order capacitors and inductors, their implementation can be performed using fractional-order differentiators and integrators, respectively, combined with a voltage-to-current conversion stage. The transfer function of fractional-order differentiators and integrators can be approximated through the utilization of appropriate integer-order transfer functions. In order to achieve that, the Continued Fraction Expansion as well as the Oustaloup’s approximations can be utilized. The accuracy, in terms of magnitude and phase response, of transfer functions of differentiators/integrators derived through the employment of the aforementioned approximations, is very important factor for achieving high performance approximation of the fractional-order elements. A comparative study of the accuracy offered by the Continued Fraction Expansion and the Oustaloup’s approximation is performed in this paper. As a next step, the corresponding implementations of the emulators of the fractional-order elements, derived using fundamental active cells such as operational amplifiers, operational transconductance amplifiers, current conveyors, and current feedback operational amplifiers realized in commercially available discrete-component IC form, are compared in terms of the most important performance characteristics. The most suitable of them are further compared using the OrCAD PSpice software.

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