On Generalized Pell Numbers of Order r ≥ 2

In this paper we investigate the generalized Pell numbers of order r ≥ 2 through the properties of their related fundamental system of generalized Pell numbers. That is, the generalized Pell number of order r ≥ 2; are expressed as a linear combination of a fundamental system of generalized Pell numbers. The properties of this fundamental system are examined and results can be established for generalized Pell numbers of order r ≥ 2. Some identities and combinatorial results are established. Moreover, the analytic study of the fundamental system of generalized Pell is provided. Furthermore, the generalized Pell Cassini identity type is provided.

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Gaussian Pell Numbers

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.26659 ◽

2018 ◽

Vol 7 (4.10) ◽

pp. 1012

Author(s):

P. Balamurugan ◽

A. Gnanam

Keyword(s):

Recurrence Relation ◽

Complex Plane ◽

Complex Numbers ◽

Two Dimensional ◽

Dimensional Approach ◽

Pell Numbers ◽

Pell Number

Gaussian numbers means representation as Complex numbers. In this work, Gaussian Pell numbers are defined from recurrence relation of Pell numbers. Here the recurrence relation on Gaussian Pell number is represented in two dimensional approach. This provides an extension of Pell numbers into the complex plane.

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Summing a Family of Generalized Pell Numbers

Annales Mathematicae Silesianae ◽

10.2478/amsil-2020-0024 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Helmut Prodinger

Keyword(s):

Linear Combination ◽

Computer Algebra ◽

Generating Functions ◽

Fibonacci Numbers ◽

Partial Sums ◽

Pell Numbers ◽

New Family ◽

Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

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Pell numbers whose Euler function is a Pell number

Publications de l Institut Mathematique ◽

10.2298/pim1715231f ◽

2017 ◽

Vol 101 (115) ◽

pp. 231-245

Author(s):

Bernadette Faye ◽

Florian Luca

Keyword(s):

Euler Function ◽

Pell Numbers ◽

Pell Number

We show that the only Pell numbers whose Euler function is also a Pell number are 1 and 2.

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There are no Diophantine quadruples of Pell numbers

International Journal of Number Theory ◽

10.1142/s179304212250004x ◽

2021 ◽

pp. 1-19

Author(s):

Salah Eddine Rihane ◽

Florian Luca ◽

Alain Togbé

Keyword(s):

Positive Integer ◽

Pell Numbers ◽

Pell Number ◽

Diophantine Quadruples ◽

Positive Integers

In this paper, we prove that there are no positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] such that [Formula: see text] is a Diophantine quadruple, where for a positive integer [Formula: see text], [Formula: see text] is the [Formula: see text]th Pell number.

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On Pillai's problem with Pell numbers and powers of 2

10.46298/hrj.2019.5102 ◽

2019 ◽

Author(s):

Mohand Ouamar Hernane ◽

Florian Luca ◽

Salah Rihane ◽

Alain Togbé

Keyword(s):

Pell Numbers ◽

Powers Of 2 ◽

International Audience ◽

Pell Number

International audience In this paper, we find all integers c having at least two representations as a difference between a Pell number and a power of 2.

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