On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci Numbers

Abstract

Let $(F_n{)}_{n\ge 0}$ and $(L_n{)}_{n\ge 0}$ be the Fibonacci and Lucas sequences, respectively. In this paper we determine all Fibonacci numbers which are mixed concatenations of a Fibonacci and a Lucas numbers. By mixed concatenations of $a$ and $b$, we mean the both concatenations $\overline{ab}$ and $\overline{ba}$ together, where $a$ and $b$ are any two non negative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations $F_n=10^d{F}_m+L_k$ and $F_n=10^d{L}_m+F_k$ in non-negative integers $(n,m,k),$ where $d$ denotes the number of digits of $L_k$ and $F_k$, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.