On bi-periodic Padovan and Perrin quaternions over finite fields

TL;DR

This paper explores bi-periodic Padovan and Perrin quaternion sequences over finite fields, analyzing their algebraic properties, norms, and conditions for zero divisors and invertibility in quaternion algebra.

Contribution

It introduces bi-periodic Perrin sequences, extends them to quaternion algebra, and characterizes algebraic properties over finite fields, which is a novel extension.

Findings

Explicit criteria for zero divisors and invertible elements.

Structural relationship between bi-periodic Perrin and Padovan sequences.

Analysis of norm properties in quaternion algebra over finite fields.

Abstract

In this paper, we investigate bi-periodic Padovan and bi-periodic Perrin quaternions over the quaternion algebra Q_Zp. We introduce the bi-periodic Perrin sequence and clarify its structural relationship with the bi-periodic Padovan sequence. By extending these sequences to the quaternion setting, we analyze their norm properties in the modular framework. For suitable choices of twin prime coefficients, we derive explicit criteria characterizing zero divisors and invertible elements in Q_Zp.