On the Diophantine equation L_n − L_m = 3 • 2^a

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
Full paper (PDF, 191 Kb)

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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 L_n − L_m = 3 • 2^a are given by L₁₁ − L₄ = 199 − 7 = 3 • 2⁶, L₄ − L₃ = 7 − 4 = 3 • 2⁰, L₄ − L₁ = 7 − 1 = 3 • 2, L₃ − L₁ = 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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