Some applications of Fibonacci and Lucas numbers

TL;DR

This paper explores new algebraic structures, generalizations, and applications of Fibonacci and Lucas numbers, including their use in constructing split quaternion algebras and analyzing their properties.

Contribution

It introduces generalized Fibonacci and Lucas sequences via arbitrary binary relations and applies these to develop new algebraic structures and quaternion algebra examples.

Findings

Algebraic structures on sets defined by Fibonacci and Lucas numbers

Generalization of Fibonacci and Lucas sequences using arbitrary binary relations

Method for constructing new split quaternion algebra examples

Abstract

In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary relation over the real fields instead of addition of the real numbers and we give properties of the new obtained sequences. Moreover, by using some relations between Fibonacci and Lucas numbers, we provide a method to find new examples of split quaternion algebras and we give new properties of these elements.