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

Abstract

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ 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.