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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## 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 *L _{n}* −

*L*= 3 • 2

_{m}^{a}are given by

*L*

_{11}−

*L*

_{4}= 199 − 7 = 3 • 2

^{6},

*L*

_{4}−

*L*

_{3}= 7 − 4 = 3 • 2

^{0},

*L*

_{4}−

*L*

_{1}= 7 − 1 = 3 • 2,

*L*

_{3}−

*L*

_{1}= 4 − 1 = 3 • 2

^{0}. 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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*cx*^{2}; Journal of Integer Sequences, 14, Article 11.9.3. - 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.
- Pink, I., & Ziegler, V. (2018) Effective resolution of Diophantine equations of the form
*u*+_{n}*u*=_{m}*wp*_{1}^{z1}*p*_{2}^{z2}…*p*; Monaths Math, 185, 103–131._{s}^{zs} - Şiar, Z., & Keskin, R. On the Diophantine equation
*F*−_{n}*F*= 2_{m}(preprint) Available online: arXiv:1712.10138v1.^{a}

## Related papers

## Cite this paper

APAŞiar, Z., & Keskin. R. (2018). On the Diophantine equation *L _{n}* −

*L*= 3 • 2

_{m}^{a}.

*Notes on Number Theory and Discrete Mathematics*, 24(4), 112-119, doi: 10.7546/nntdm.2018.24.4.112-119.

Şiar, Ziar and Refik Keskin. “On the Diophantine Equation *L _{n}* −

*L*= 3 • 2

_{m}^{a}.” Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 112-119, doi: 10.7546/nntdm.2018.24.4.112-119.

Şiar, Ziar and Refik Keskin. “On the Diophantine Equation *L _{n}* −

*L*= 3 • 2

_{m}^{a}.” Notes on Number Theory and Discrete Mathematics 24.4 (2018): 112-119. Print, doi: 10.7546/nntdm.2018.24.4.112-119.