Fibonacci numbers as mixed concatenations of Fibonacci and Lucas numbers (Q6563198)

Let \( (F_n)_{n\ge 0} \) and \( (L_n)_{n\ge 0} \) be the sequences of Fibonacci and Lucas numbers, respectively, defined by the linear recurrence relations: \( F_0=0 \), \( F_1=1 \), and \( F_n=F_{n-1}+F_{n-2} \) for all \( n\ge 2 \); and \( L_0=2, L_1= 1\), and \( L_n=L_{n-1}+L_{n=2} \) for all \( n\ge 2 \). In the paper under review, the authors study the Diophantine equations: \N\begin{align*}\NF_n=10^{d}F_m+L_k \quad \text{and} \quad F_n=10^{d}L_m+F_k,\tag{1}\N\end{align*}\Nin nonnegative integers \( n,m,k \), and where \( d \) represents the number of digits of \( L_k \) and \( F_k \) in base \( 10 \), respectively. In their main results, the authors show that: all Fibonacci numbers which are concatenations of a Fibonacci and a Lucas numbers are only the numbers \(1, 2, 3, 13, 21\), and \(34\); and all Fibonacci numbers which are concatenations of a Lucas and a Fibonacci numbers are only the numbers \(13\) and \(21\).\N\NThe proofs of their results follow from a clever combination of techniques in Diophantine number theory, the usual properties of the Fibonacci and Lucas sequences, Baker's theory for non-zero lower bounds for linear forms in logarithms of algebraic numbers, and reduction techniques involving the theory of continued fractions. All computations are done with the aid of a simple computer program in \texttt{Maple}.