On Companion sequences associated with Leonardo quaternions: Applications over finite fields

TL;DR

This paper introduces Lucas-Leonardo p-quaternions, explores their properties, and investigates applications of companion sequences related to Leonardo quaternions, including identifying special elements over finite fields.

Contribution

It presents a new class of quaternions called Lucas-Leonardo p-quaternions and studies their properties and applications in finite fields.

Findings

Identification of zero divisors and invertible elements in quaternion algebras over finite fields

Derivation of fundamental properties of Lucas-Leonardo p-quaternions

Application of companion sequences to analyze quaternion elements

Abstract

It is known that the quaternion algebras are central simple algebras and also clifford algebras. In this paper, we introduce a new class of quaternions called Lucas-Leonardo p-quaternions and derive several fundamental properties of these numbers. Furthermore, we investigate some applications related to companion sequences associated with Leonardo quaternions. In particular, we determine Lucas-Leonardo quaternions and Francois quaternions, which are zero divisors and invertible elements in the quaternion algebra over certain finite fields.