On Companion sequences associated with Leonardo quaternions: Applications over finite fields
Diana Savin, Elif Tan
TL;DR
The paper introduces a new class of quaternions, the Lucas-Leonardo p-quaternions, built from Lucas-Leonardo p-numbers and embedded in the real quaternion algebra with i^2 = j^2 = k^2 = ijk = -1. It derives their fundamental properties, including recurrences, a generating function, and relations to Fibonacci/Lucas/Leonardo companions, and then extends the study to quaternion algebras over finite fields to identify zero-divisors and units for Lucas-Leonardo and Francois quaternions. It provides explicit modular criteria for q = 3, 5, 7 that characterize invertibility versus zero divisors via norms, and shows how setting p = 1 reduces to Lucas-Leonardo quaternions. The work connects sequence theory with quaternion algebra over finite fields, offering concrete tools for applications in coding theory, cryptography, and algebraic number theory.
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.