Mulatu Numbers Which Are Concatenation of Two Fibonacci Numbers

Year 2023, , 1122 - 1127, 18.10.2023

Fatih Erduvan , Merve Güney Duman

https://doi.org/10.16984/saufenbilder.1235571

Abstract

Let (M_k) be the sequence of Mulatu numbers defined by M_0=4, M_1=1, M_k=M_(k-1)+M_(k-2) and (F_k) be the Fibonacci sequence given by the recurrence F_k=F_(k-1)+F_(k-2) with the initial conditions F_0=0, F_1=1 for k≥2. In this paper, we showed that all Mulatu numbers, that are concatenations of two Fibonacci numbers are 11,28. That is, we solved the equation M_k=〖10〗^d F_m+F_n, where d indicates the number of digits of F_n. We found the solutions of this equation as (k,m,n,d)∈{(4,2,2,1),(6,3,6,1)}. Moreover the solutions of this equation displayed as M_4=(F_2 F_2 ) ̅=11 and M_6=(F_3 F_6 ) ̅=28. Here the main tools are linear forms in logarithms and Baker Davenport basis reduction method.

Keywords

Mulatu and Fibonacci numbers, linear forms in logarithms, exponential Diophantine equations

References

• M. Lemma, ‘‘The Mulatu Numbers’’ Advances and Applications in Mathematical Sciences, vol. 10, no. 4, pp. 431-440, 2011.
• W.D. Banks, F. Luca, ‘‘Concatenations with binary recurrent sequences’’ Journal of Integer Sequences, vol. 8, no. 5, pp. 1-3, 2005.
• M. Alan, ‘‘On Concatenations of Fibonacci and Lucas Numbers’’ Bulletin of the Iranian Mathematical Society, vol. 48, no. 5, pp. 2725-2741, 2022.
• M. Lemma, J. Lambrigt, “Some Fascinating theorems of Mulatu Numbers”, Hawai University International Conference, 2016.
• N. Irmak, Z. Siar, R. Keskin, “On the sum of three arbitrary Fibonacci and Lucas numbers” Notes on Number Theory and Discrete Mathematics, vol. 25, no. 4, pp. 96-101, 2019.
• Y. Bugeaud, ‘‘Linear Forms in Logarithms and Applications’’ IRMA Lectures in Mathematics and Theoretical Physics 28, Zurich, European Mathematical Society, 1-176, 2018.
• Y. Bugeaud, M. Mignotte S. Siksek, ‘‘Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers’’ Annals of Mathematics, vol. 163, no. 3, pp. 969-1018, 2006.
• J.J. Bravo, C.A. Gomez, F. Luca, ‘‘Powers of two as sums of two k-Fibonacci numbers’’ Miskolc Mathematical Notes, vol. 17, no. 1, pp. 85-100, 2016.
• A. Dujella, A. Pethò, ‘‘A generalization of a theorem of Baker and Davenport’’ Quarterly Journal of Mathematics Oxford series (2), vol. 49, no. 3, pp. 291-306, 1998.
• B. M. M. de Weger, ‘‘Algorithms for Diophantine Equations’’ CWI Tracts 65, Stichting Mathematisch Centrum, Amsterdam, 1-69, 1989.