On bi-periodic Padovan and Perrin quaternions over finite fields

TL;DR

The paper introduces bi-periodic Padovan and Perrin sequences and their quaternion extensions over the split quaternion algebra $Q_{\mathbb{Z}_p}$ using twin-prime coefficients $a=p-2$, $b=p$, and establishes explicit criteria for zero divisors and invertibility of the associated quaternions. By deriving parity-dependent recurrences, generating functions, and modular reductions, the authors connect quaternion norms to Fibonacci numbers via $P_{2k} \equiv (-1)^k (F_{k+3}-1) \pmod{p}$ and related relations, leveraging Fibonacci entry points $z(p)$ and Pisano periods $\pi(p)$. They provide complete zero-divisor characterizations for bi-periodic Padovan quaternions and analogous results for bi-periodic Perrin quaternions, including special-case analyses for primes $p=7,13,181$ (notably invertibility for all odd-index Perrin quaternions at $p=13$). The work reveals deep links between higher-order periodic recurrences, quaternion algebras over finite fields, and number-theoretic invariants, offering a general framework with concrete implications and directions for higher-dimensional $k$-periodic extensions.

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.

Theorems & Definitions (25)

• Definition 1
• Lemma 1
• proof
• Proposition 1
• proof
• Lemma 2
• Theorem 1
• proof
• Theorem 2
• proof