Nurettin Irmak and Alain Togbé

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 24, 2018, Number 3, Pages 42—49

DOI: 10.7546/nntdm.2018.24.3.95-102

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## Details

### Authors and affiliations

Nurettin Irmak

*Mathematics Department, Art and Science Faculty
Nigde Ömer Halisdemir University, Nigde, Turkey
*

Alain Togbé

*Department of Mathematics, Purdue University Northwest
1401 S. U. S. 421., Westville, IN 46391, United States
*

### Abstract

Let (𝐿^{𝑛})^{𝑛≥0} be the Lucas sequence. D. Marques and A. Togbé [7] showed that if 𝐹_{𝑛}…𝐹_{𝑛+𝑘−1} is a repdigit with at least two digits, then (𝑘, 𝑛) = (1, 10), where (𝐹_{𝑛})_{≥0} is the Fibonacci sequence. In this paper, we solve the equation 𝐿_{𝑛}…𝐿_{𝑛+𝑘−1} = 𝑎 (︂10^{𝑚} − 1) / 9, where 1 ≤ 𝑎 ≤ 9, 𝑛, 𝑘 ≥ 2 and 𝑚 are positive integers.

### Keywords

- Lucas numbers
- Repdigits

### 2010 Mathematics Subject Classification

- 11A63
- 11B39
- 11B50

### References

- Bugeaud, Y., Mignotte, M., & Siksek, S. (2006) Classical and modular approaches to exponential Diophantine equations, I. Fibonacci and Lucas powers, Ann. of Math., 163, 969–1018.
- Dujella, A., A. Pethö (1998) A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49 (195), 291–306.
- Koshy, T. (2003) Fibonacci and Lucas Numbers with Applications, Wiley.
- Lengyel, T. (1995) The order of the Fibonacci and Lucas numbers, Fibonacci Quart., 33(3), 234–239.
- Luca, F. (2000) Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math. 50, 243–254.
- Luca, F., T. N. Shorey (2005) Diophantine equations with product of consecutive terms in Lucas sequences, J. Number Theory, 114, 298–311.
- Marques, D., & Togbé, A. (2012) On repdigits as product of consecutive Fibonacci numbers. Rend. Istit. Mat. Univ. Trieste, 44, 393–397.
- Matveev, E. M. (2000) An explicit lower bound for a homogeneous linear form in logarithms of algebraic numbers. II, Izv. Ross. Akad. Nauk Ser. Mat. 64(6), 125–180; translation in Izv. Math. 64 (200), 6, 1217–1269.

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## Cite this paper

APAIrmak, N., & Togbé, A. (2018). On repdigits as product of consecutive Lucas numbers, Notes on Number Theory and Discrete Mathematics, 24(3), 95-102, doi: 10.7546/nntdm.2018.24.3.95-102.

ChicagoIrmak, Nurettin, and Alain Togbé. “On Repdigits as Product of Consecutive Lucas Numbers.” Notes on Number Theory and Discrete Mathematics 24, no. 3 (2018): 95-102, doi: 10.7546/nntdm.2018.24.3.95-102.

MLAIrmak, Nurettin, and Alain Togbé. “On Repdigits as Product of Consecutive Lucas Numbers.” Notes on Number Theory and Discrete Mathematics 24.3 (2018): 95-102. Print, doi: 10.7546/nntdm.2018.24.3.95-102.