Markov equation with components of some binary recurrent sequences

S. G. Rayaguru, M. K. Sahukar and G. K. Panda
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 3, Pages 149–159
DOI: 10.7546/nntdm.2020.26.3.149-159
Full paper (PDF, 324 Kb)

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Authors and affiliations

S. G. Rayaguru
Department of Mathematics, National Institute of Technology
Rourkela, India

M. K. Sahukar
Department of Mathematics, National Institute of Technology
Rourkela, India

G. K. Panda
Department of Mathematics, National Institute of Technology
Rourkela, India

Abstract

The generalized Lucas sequence {U_n}_n≥0 is defined by U_n+1 = rU_n + sU_n−1; n ≥ 0 with U₀ = 0; U₁ = 1 of which the Fibonacci sequence (F_n) is the particular case r = s = 1. In 2018, F. Luca and A. Srinivasan searched for the solutions x, y, z ∈ F_n of the Markov equation x² + y² + z² = 3xyz and proved that (F₁; F_2n−1, F_2n+1); n ≥ 1 is the only solution. In this paper, we extend this work from the Fibonacci sequence to any generalized Lucas sequence U_n for the case s = ±1.

Keywords

• Lucas sequences
• Markov equation
• Markov triples

2010 Mathematics Subject Classification

• 11B39
• 11D99

References

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