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
Full paper (PDF, 193 Kb)

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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 (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

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