On the x-coordinates of Pell equations that are products of two Pell numbers
-
Mahadi Ddamulira
and
Florian Luca
Abstract
Let (Pm)m≥0 be the sequence of Pell numbers given by P0 = 0, P1 = 1, and Pm+2 = 2Pm+1 + Pm for all m ≥ 0. In this paper, for an integer d ≥ 2 which is square free, we show that there is at most one value of the positive integer x participating in the Pell equation x2 − dy2 = ± 1, which is a product of two Pell numbers.
This work has been supported by the Mathematics for Sustainable Development (MATH4SDG) project, which is a research and development project running in the period 2021-2026 at Makerere University, Uganda, University of Dar es Salaam, Tanzania, and the University of Bergen, Norway, funded through the NORHED II programme under the Norwegian Agency for Development Cooperation (NORAD, project no 68105).
(Communicated by István Gaál)
Acknowledgement
The authors thank the referee for the careful reading of the manuscript and the comments that have greatly improved the current version of this paper. Part of this work was done when both authors visited the Max Planck Institute for Mathematics in Bonn in the Fall of 2019. They thank this institution for hospitality, support and excellent working conditions.
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- Large deviations for some dependent heavy tailed random sequences
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- In memory of Paolo
Articles in the same Issue
- Prof. RNDr. Gejza Wimmer, DrSc. – 3/4 C?
- On hyper (r, q)-Fibonacci polynomials
- The pointfree version of 𝓒c(X) via the ranges of functions
- On the x-coordinates of Pell equations that are products of two Pell numbers
- Subordination properties and coefficient problems for a novel class of convex functions
- Certain radii problems for 𝓢∗(ψ) and special functions
- On direct and inverse Poletsky inequalities with a tangential dilatation
- Fourth-order nonlinear strongly non-canonical delay differential equations: new oscillation criteria via canonical transform
- Hermite interpolation of type total degree associated with certain spaces of polynomials
- Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem
- A fixed point technique to the stability of Hadamard 𝔇-hom-der in Banach algebras
- Compact subsets of Cλ,u(X)
- Variations of star selection principles on hyperspaces
- On lower density operators
- Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠2t,3
- On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence
- Large deviations for some dependent heavy tailed random sequences
- Metric, stratifiable and uniform spaces of G-permutation degree
- In memory of Paolo
