# On the Diophantine equation Ln − Lm = 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

## Details

### 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

• 11B39
• 11D61
• 11J86

### References

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11. Şiar, Z., & Keskin, R. On the Diophantine equation FnFm = 2a (preprint) Available online: arXiv:1712.10138v1.

## 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.