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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## Details

### 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

Rourkela, India

G. K. Panda

*Department of Mathematics, National Institute of Technology*

Rourkela, India

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

*U*

_{0}= 0;

*U*

_{1}= 1 of which the Fibonacci sequence (

*F*) is the particular case

_{n}*r*=

*s*= 1. In 2018, F. Luca and A. Srinivasan searched for the solutions

*x, y, z*∈

*F*of the Markov equation

_{n}*x*

^{2}+

*y*

^{2}+

*z*

^{2}= 3

*xyz*and proved that (

*F*

_{1};

*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*for the case

_{n}*s*= ±1.

### Keywords

- Lucas sequences
- Markov equation
- Markov triples

### 2010 Mathematics Subject Classification

- 11B39
- 11D99

### References

- Behera, A., & Panda, G. K. (1999). On the square roots of triangular numbers, Fib. Quart., 37 (2), 98-105.
- Bolat, C., & Köse, H. (2010). On the Properties of
*k*-Fibonacci Numbers, Int. J. Contemp. Math. Sciences, 5 (22), 1097–1105. - Luca, F., & Srinivasan, A. (2018). Markov equation with Fibonacci components, The Fibonacci Quarterly, 56 (2), 126–129.
- Markoff, A. (1880). Sur les formes quadratiques binaires indéfinies, Math. Ann., 17 (3), 379–399.
- 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.
- 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.