Mulatu Numbers That Are Concatenations of Two Lucas Numbers

Authors : Fatih ERDUVAN

Pages : 914-920

Doi:10.35414/akufemubid.1240679

View : 44 | Download : 56

Publication Date : 2023-08-31

Article Type : Research Paper

Abstract :In this paper, we find that all Mulatu numbers, which are concatenations of two Lucas numbers are 11,17,73,118. Let 〖insert ignore into journalissuearticles values(M_k);〗_insert ignore into journalissuearticles values(k≥0); and 〖insert ignore into journalissuearticles values(L_k);〗_insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(k,m,n,d);, where d denotes the number of digits of L_n. Solutions of this equation are denoted by insert ignore into journalissuearticles values(k,m,n,d);=insert ignore into journalissuearticles values(4,1,1,1); insert ignore into journalissuearticles values(5,1,4,1); insert ignore into journalissuearticles values(8,4,2,1); insert ignore into journalissuearticles values(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.
Keywords : Lucas sayıları, Mulatu sayıları, Logaritmalarda lineer formlar, Diophantine denklemleri