Rad HAZU, Matematičke znanosti, Vol. 29 (2025), 63-82.

A GENERALIZATION OF A THEOREM OF MURAT ALAN

Mariama Ndao Faye, Kouèssi Norbert Adédji and Alain Togbé

UFR of Applied Sciences and Technology, University Gaston Berger, Sénégal
e-mail: fayemariamandao@gmail.com

Institut de Mathématiques et de Sciences Physiques, Université d'Abomey-Calavi, Bénin
e-mail: adedjnorb1988@gmail.com

Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street, Hammond, IN 46323 USA
e-mail: atogbe@pnw.edu

Abstract.   Let (Fn)n≥0 and (Ln)n≥0 be the Fibonacci and Lucas sequences respectively. In 2022, Murat Alan found all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. Let b ≥ 2 be an integer. In this paper, we generalize the results of Murat Alan by considering the following Diophantine equations Fn = bdLm + Lk and Ln = bdFm + Fk in non-negative integers (n, m, k), where d denotes the number of digits of Lk and Fk in base b, respectively.

2020 Mathematics Subject Classification.   11B36, 11J68, 11J86.

Key words and phrases.   Fibonacci numbers, Lucas numbers, b-concatenation, logarithmic height, reduction method.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/yl4okf8no9

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