On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci Numbers

TL;DR

This paper characterizes all Fibonacci numbers that can be formed by concatenating a Fibonacci and a Lucas number in either order, using advanced number theory techniques.

Contribution

It provides a complete solution to the problem of identifying Fibonacci numbers that are mixed concatenations of Fibonacci and Lucas numbers, solving associated Diophantine equations.

Findings

Identifies all Fibonacci numbers that are concatenations of Fibonacci and Lucas numbers.

Uses bounds for linear forms in logarithms and reduction methods to solve the equations.

Establishes the specific solutions satisfying the concatenation conditions.

Abstract

Let $(F_n)_{n\geq 0}$ and $(L_n)_{n\geq 0}$ 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 $\overline{ab}$ and $\overline{ba}$ together, where $ a $ and $ b $ are any two non negative 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 non-negative 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.