Diophantine equations with Lucas and Fibonacci number coefficients

Cemil Karaçam, Alper Vural, Bilal Aytepe, and Ferhat Yıldız
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 181–190
DOI: 10.7546/nntdm.2025.31.1.181-190
Full paper (PDF, 239 Kb)

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Authors and affiliations

Cemil Karaçam
Department of Mathematics, Muğla Science and Art Center
Muğla, Türkiye

Alper Vural
Department of Computer Engineering, Boğaziçi University
Istanbul, Türkiye

Bilal Aytepe
Department of Mathematical Engineering, Yıldız Technical University
Istanbul, Türkiye

Ferhat Yıldız
Department of Economics, Boğaziçi University
Istanbul, Türkiye

Abstract

Fibonacci and Lucas numbers are special number sequences that have been the subject of many studies throughout history due to the relations they provide. The studies are continuing today, and findings about these number sequences are constantly increasing. The relations between the Fibonacci and Lucas numbers, which were found during the proof of the prime between two consecutive numbers belonging to the Fibonacci or Lucas number sequence with the Euclidean algorithm, started our project. In the project, Diophantine equations whose coefficients are Lucas or Fibonacci numbers have been studied, various relations have been found, and their proofs have been made.

   

As in the above example, the equivalents of and values were found in the Diophantine equations with Fibonacci and Lucas number coefficients; and based on this example, different variations of the Diophantine equations whose coefficients were selected from the Fibonacci and Lucas number sequences were created, and their proofs were made.

Secondly, the geometric shapes consisting of vertices determined by pair of numbers selected from the Fibonacci or Lucas number sequence were considered, and their properties were examined. Various relations were found between them, and generalizations were made.

Keywords

• Fibonacci numbers
• Diophantine equations
• Polygon areas

2020 Mathematics Subject Classification

• 11B39
• 11D04

References

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Manuscript history

• Received: 19 March 2024
• Revised: 13 April 2025
• Accepted: 14 April 2025
• Online First: 30 April 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).