On repdigits as product of consecutive Lucas numbers

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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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 (𝐿𝑛)𝑛≥0 be the Lucas sequence. D. Marques and A. Togbé [7] showed that if 𝐹𝑛…𝐹𝑛+𝑘−1 is a repdigit with at least two digits, then (𝑘, 𝑛) = (1, 10), where (𝐹𝑛)≥0 is the Fibonacci sequence. In this paper, we solve the equation 𝐿𝑛…𝐿𝑛+𝑘−1 = 𝑎 (︂10𝑚 − 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

  1. 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.
  2. Dujella, A., A. Pethö (1998) A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49 (195), 291–306.
  3. Koshy, T. (2003) Fibonacci and Lucas Numbers with Applications, Wiley.
  4. Lengyel, T. (1995) The order of the Fibonacci and Lucas numbers, Fibonacci Quart., 33(3), 234–239.
  5. Luca, F. (2000) Fibonacci and Lucas numbers with only one distinct digit, Portugal. Math. 50, 243–254.
  6. Luca, F., T. N. Shorey (2005) Diophantine equations with product of consecutive terms in Lucas sequences, J. Number Theory, 114, 298–311.
  7. Marques, D., & Togbé, A. (2012) On repdigits as product of consecutive Fibonacci numbers. Rend. Istit. Mat. Univ. Trieste, 44, 393–397.
  8. 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

APA

Irmak, 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.

Chicago

Irmak, 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.

MLA

Irmak, 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.

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