Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers

Abstract

Let (F-n)(n >= 0) and (L-n)(n >= 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 (ab) over bar and (ba) over bar together, where a and b are any two nonnegative 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 nonnegative 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.