On Mixed Concatenations of Fibonacci and Lucas Numbers Which are Fibonacci Numbers
Alaa Altassan, Murat Alan

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 and 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 and , we mean the both concatenations and together, where and are any two non negative integers. So, the mathematical formulation of this problem leads us searching the solutions of two Diophantine equations and in non-negative integers where denotes the number of digits of and , respectively. We use lower bounds for linear forms in logarithms and reduction method in Diophantine approximation to get the results.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Identities · Analytic Number Theory Research
