On mixed $b$-concatenations of Fibonacci and Lucas numbers that are Lucas numbers

TL;DR

This paper investigates which Lucas numbers can be expressed as mixed base-$b$ concatenations of Fibonacci and Lucas numbers, proving that only finitely many such numbers exist through advanced Diophantine approximation techniques.

Contribution

It characterizes all Lucas numbers that are base-$b$ concatenations of Fibonacci and Lucas numbers, establishing finiteness results using Diophantine methods.

Findings

Only finitely many Lucas numbers are such concatenations.

The paper provides explicit bounds and conditions for these representations.

It combines Diophantine approximation and continued fractions techniques.

Abstract

Let $(F_n)_{n\ge0}$ and $(L_n)_{n\ge0}$ denote the sequences of Fibonacci and Lucas numbers respectively. This paper determines all Lucas numbers that can be represented as base $b$ mixed concatenations of a Fibonacci number and a Lucas number. Mathematically, we study of two Diophantine equations $L_n=b^dL_m+F_k$ and $L_n=b^dF_m+L_k$, where $d$ is the number of digits of $F_k$ or $L_k$ in base $b$. To tackle these equations, we combine tools from Diophantine approximation on non-zero linear forms in logarithms and reduction methods based on continued fractions. This allows us to prove that only finitely many such Lucas numbers exist.