Mulatu Numbers Which Are Concatenation of Two Fibonacci Numbers

Anahtar Kelimeler:

Linear forms in logarithms, Mulatu and Fibonacci numbers, reduction method, exponantial Diophantine equations

Mulatu Numbers Which Are Concatenation of Two Fibonacci Numbers

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,

PDF

___

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.