Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers

Abstract

AbstractLet (Fn)n≥0and (Ln)n≥0be 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 ofaandb, we mean the both concatenationsabandbatogether, whereaandbare any two nonnegative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equationsFn= 10dFm+LkandFn= 10dLm+Fkin nonnegative integers (n,m,k), whereddenotes the number of digits ofLkandFk, respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.