On Concatenations of Fibonacci and Lucas Numbers

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ be the Fibonacci and Lucas sequences. In this paper we determine all Fibonacci and Lucas numbers which are concatenations of two terms of the other sequence. This problem is identical to solve the Diophantine equations $ F_n=10^d L_m +L_k $ and $ L_n=10^d F_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.