Abstract
Abstract Let (Fn ) n≥0 and (Ln ) 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 and ba 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 Fn = 10 d Fm + Lk and Fn = 10 d Lm + Fk in nonnegative integers (n, m, k), where d denotes the number of digits of Lk and Fk , respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
Published Version
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