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Mulatu Numbers That Are Concatenations of Two Lucas Numbers

In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118. Let 〖(M_k)〗_(k≥0) and 〖(L_k)〗_(k≥0) be the Mulatu and Lucas sequences. That is, we solve the Diophantine equation M_k=L_m L_n=10^d L_m+L_n in non-negative integers (k,m,n,d), where d denotes the number of digits of L_n. Solutions of this equation are denoted by (k,m,n,d)=(4,1,1,1),(5,1,4,1),(8,4,2,1),(9,1,6,2). In other words, we have the solutions M_4=L_1 L_1=11, M_5=L_1 L_4=17, M_8=L_4 L_2=73, M_9=L_1 L_6=118. The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.

Afyon Kocatepe Universitesi Fen Ve Muhendislik Bilimleri Dergisi

Fatih ERDUVAN

Afyon Kocatepe University Journal of Sciences and Engineering

2023

Title: Mulatu Numbers That Are Concatenations of Two Lucas Numbers

Description:

In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118.

Let 〖(M_k)〗_(k≥0) and 〖(L_k)〗_(k≥0) be the Mulatu and Lucas sequences.

That is, we solve the Diophantine equation M_k=L_m L_n=10^d L_m+L_n in non-negative integers (k,m,n,d), where d denotes the number of digits of L_n.

Solutions of this equation are denoted by (k,m,n,d)=(4,1,1,1),(5,1,4,1),(8,4,2,1),(9,1,6,2).

In other words, we have the solutions M_4=L_1 L_1=11, M_5=L_1 L_4=17, M_8=L_4 L_2=73, M_9=L_1 L_6=118.

The proof based on Baker’s theory and we used linear forms in logarithms and reduction method to solve of this Diophantine equation.

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Mulatu Numbers That Are Concatenations of Two Lucas Numbers

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