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
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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 {Un}n≥0 is defined by Un+1 = rUn + sUn−1; n ≥ 0 with U0 = 0; U1 = 1 of which the Fibonacci sequence (Fn) is the particular case r = s = 1. In 2018, F. Luca and A. Srinivasan searched for the solutions x, y, zFn of the Markov equation x2 + y2 + z2 = 3xyz and proved that (F1; F2n−1, F2n+1); n ≥ 1 is the only solution. In this paper, we extend this work from the Fibonacci sequence to any generalized Lucas sequence Un for the case s = ±1.

Keywords

  • Lucas sequences
  • Markov equation
  • Markov triples

2010 Mathematics Subject Classification

  • 11B39
  • 11D99

References

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  3. Luca, F., & Srinivasan, A. (2018). Markov equation with Fibonacci components, The Fibonacci Quarterly, 56 (2), 126–129.
  4. Markoff, A. (1880). Sur les formes quadratiques binaires indéfinies, Math. Ann., 17 (3), 379–399.
  5. Panda, G. K., & Ray, P. K. (2011). Some Links of Balancing and Cobalancing Numbers with Pell and Associated Pell Numbers, Bull. Inst. Math., Acad. Sin., 6 (1), 41–72.
  6. Sloane, N. J. A. On-Line Encyclopedia of Integer Sequences, Available online at: https://oeis.org/.

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Cite this paper

Rayaguru, S. G., Sahukar, M. K., & Panda, G. K. (2020). Markov equation with components of some binary recurrent sequences. Notes on Number Theory and Discrete Mathematics, 26 (3), 149-159, doi: 10.7546/nntdm.2020.26.3.149-159.

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