On the x-coordinates of Pell equations that are products of two Padovan numbers

Mahadi Ddamulira

On the x-coordinates of Pell equations that are products of two Padovan numbers

Mahadi Ddamulira^*

^*Corresponding author for this work

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

Let (P _n) _n≥0 be the sequence of Padovan numbers defined by P ₀ = 0, P ₁ = P ₂ = 1, and P _n+3 = P _n+1 + P _n for all n ≥ 0. In this paper, we find all positive square-free integers d ≥ 2 such that the Pell equations x ² − dy ² = ℓ, where ℓ ∈ {±1, ±4}, have at least two positive integer solutions (x, y) and (x ^′, y ^′) such that each of x and x ^′ is a product of two Padovan numbers.

Original language	English
Article number	A70
Pages (from-to)	1-20
Number of pages	20
Journal	INTEGERS: Electronic Journal of Combinatorial Number Theory
Volume	20
Publication status	Published - 31 Aug 2020

Keywords

Padovan number
Pell equation
linear form in logarithms
Reduction method

ASJC Scopus subject areas

Algebra and Number Theory

Fields of Expertise

Information, Communication & Computing

Access to Document

http://math.colgate.edu/~integers/u70/u70.pdfLicence: CC BY 4.0

Cite this

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abstract = "Let (P n) n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = P 2 = 1, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we find all positive square-free integers d ≥ 2 such that the Pell equations x 2 − dy 2 = ℓ, where ℓ ∈ {±1, ±4}, have at least two positive integer solutions (x, y) and (x ′, y ′) such that each of x and x ′ is a product of two Padovan numbers. ",

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N2 - Let (P n) n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = P 2 = 1, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we find all positive square-free integers d ≥ 2 such that the Pell equations x 2 − dy 2 = ℓ, where ℓ ∈ {±1, ±4}, have at least two positive integer solutions (x, y) and (x ′, y ′) such that each of x and x ′ is a product of two Padovan numbers.

AB - Let (P n) n≥0 be the sequence of Padovan numbers defined by P 0 = 0, P 1 = P 2 = 1, and P n+3 = P n+1 + P n for all n ≥ 0. In this paper, we find all positive square-free integers d ≥ 2 such that the Pell equations x 2 − dy 2 = ℓ, where ℓ ∈ {±1, ±4}, have at least two positive integer solutions (x, y) and (x ′, y ′) such that each of x and x ′ is a product of two Padovan numbers.

KW - Padovan number

KW - Pell equation

KW - linear form in logarithms

KW - Reduction method

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JO - INTEGERS: Electronic Journal of Combinatorial Number Theory

JF - INTEGERS: Electronic Journal of Combinatorial Number Theory

M1 - A70

ER -

On the x-coordinates of Pell equations that are products of two Padovan numbers

Abstract

Keywords

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