On the Diophantine equation LnLm = 3 • 2a

Zafer Şiar and Refik Keskin
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 4, Pages 112—119
DOI: 10.7546/nntdm.2018.24.4.112-119
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Authors and affiliations

Zafer Şiar
Department of Mathematics, Bingöl University
Bingöl, Turkey

Refik Keskin
Department of Mathematics, Sakarya University
Sakarya, Turkey

Abstract

In this paper, we solve Diophantine equation in the tittle in positive integers m, n and a. It is shown that solutions of the equation LnLm = 3 • 2a are given by L11L4 = 199 − 7 = 3 • 26, L4L3 = 7 − 4 = 3 • 20, L4L1 = 7 − 1 = 3 • 2, L3L1 = 4 − 1 = 3 • 20. In order to prove our result, we use lower bounds for linear forms in logarithms and a version of the Baker–Davenport reduction method in Diophantine approximation.

Keywords

  • Fibonacci numbers
  • Lucas numbers
  • Exponential equations
  • Linear forms in logarithms
  • Baker’s method

2010 Mathematics Subject Classification

  • 11B39
  • 11D61
  • 11J86

References

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  9. Matveev, E. M. (2000) An Explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers II, Izv. Ross. Akad. Nauk Ser. Mat., 646, 125–180 (Russian). Translation in Izv. Math., 64, 6 (2000), 1217–1269.
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Cite this paper

APA

Şiar, Z., & Keskin. R. (2018). On the Diophantine equation LnLm = 3 • 2a. Notes on Number Theory and Discrete Mathematics, 24(4), 112-119, doi: 10.7546/nntdm.2018.24.4.112-119.

Chicago

Şiar, Ziar and Refik Keskin. “On the Diophantine Equation LnLm = 3 • 2a.” Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 112-119, doi: 10.7546/nntdm.2018.24.4.112-119.

MLA

Şiar, Ziar and Refik Keskin. “On the Diophantine Equation LnLm = 3 • 2a.” Notes on Number Theory and Discrete Mathematics 24.4 (2018): 112-119. Print, doi: 10.7546/nntdm.2018.24.4.112-119.

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