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

1. Baker, A., & Davenport, H. (1969) The equations 3x2 − 2 = y2 and 8x2 − 7 = z2; Quart. J. Math. Oxford Ser. (2), 20 (1), 129–137.
2. Bravo, E. F., & Bravo, J. J. (2015) Powers of two as sums of three Fibonacci numbers Lithuanian Mathematical Journal, 55 (3), 301–311.
3. Bravo, J. J., & Luca, F. (2014) Powers of Two as Sums of Two Lucas Numbers, Journal of Integer Sequences, 17, Article 14.8.3.
4. Bravo, J. J., & Luca, F. (2016) On the Diophantine Equation Fn + Fm = 2a, Quaestiones Mathematicae, 39 (3), 391–400.
5. Bugeaud, Y., Mignotte, M., & Siksek, S. (2006) Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers, Ann. of Math., 163(3), 969–1018.
6. Debnath, L. (2011) A short history of the Fibonacci and golden numbers with their applications, International Journal of Mathematical Education in Science and Technology, 42 (3), 337–367.
7. Dujella, A., & Pethò, A. (1998) A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2), 49 (3), 291–306.
8. Keskin, R., & Yosma, Z. (2011) On Fibonacci and Lucas Numbers of the Form cx2; Journal of Integer Sequences, 14, Article 11.9.3.
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.
10. Pink, I., & Ziegler, V. (2018) Effective resolution of Diophantine equations of the form un + um = wp1z1p2z2pszs; Monaths Math, 185, 103–131.
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.