What question did this study set out to answer?

The central aim is to explore the growth of multiplicative order in Padovan companion matrices over modular rings.

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April 23, 2026Open Access

On the Distribution of Orders of Companion Matrices over Finite Fields

Key Points

The central aim is to explore the growth of multiplicative order in Padovan companion matrices over modular rings.
Analyze the Padovan companion matrix over Z/p^eZ
Connect the behavior of orders under p-adic lifting to linear recurrences and p-adic valuation
Conduct numerical experiments to support theoretical findings.
A structural relation between base order and higher prime power extensions was formulated.
The analysis reveals a lifting phenomenon within the algebraic structure of GL(3, Z/p^eZ).
Numerical experiments confirm the proposed order growth law.

Abstract

We study the arithmetic dynamics of the Padovan companion matrix over modular rings Z/pᵉZ. We investigate the behavior of the multiplicative order under p-adic lifting and formulate a structural relation between base order and higher prime power extensions. The analysis connects linear recurrences, matrix dynamics, and p-adic valuation theory, highlighting a lifting phenomenon governed by the algebraic structure of GL (3, Z/pᵉZ). Numerical experiments support the proposed order growth law. No claim of full generality or complete proof is made. Citation: Franesi, P. (2026). p-adic Padovan Dynamics and Companion Matrix Orders Modulo Prime Powers. Zenodo. https: //doi. org/10. 5281/zenodo. 19679124

On the Distribution of Orders of Companion Matrices over Finite Fields

Key Points

Abstract

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