"On Matrix Order Growth Modulo Prime Powers: The Padovan Case"

We determine the exact order of the companion matrix associated with the Padovan recurrence modulo prime powers. In particular, we prove that the matrix has exact order 39 in GL₃(ℤ/9ℤ). The argument combines irreducibility over F₃, identification with the finite field F₂₇, and a first-order lifting mechanism. We further establish a general theorem describing how the order grows under lifting from modulo p to pᵏ. Keywords: Padovan sequence, finite fields, matrix order, lifting modulo prime powers

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-04-15
DOI
https://doi.org/10.5281/zenodo.19582696
Primary Topic
Coding theory and cryptography
Type
article
Field-Weighted Citation Impact
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article

"On Matrix Order Growth Modulo Prime Powers: The Padovan Case"

Pietro Franesi
Zenodo (CERN European Organization for Nuclear Research)
Coding theory and cryptography
article

"On Matrix Order Growth Modulo Prime Powers: The Padovan Case"

Pietro Franesi
article en

Abstract

We determine the exact order of the companion matrix associated with the Padovan recurrence modulo prime powers. In particular, we prove that the matrix has exact order 39 in GL₃(ℤ/9ℤ). The argument combines irreducibility over F₃, identification with the finite field F₂₇, and a first-order lifting mechanism. We further establish a general theorem describing how the order grows under lifting from modulo p to pᵏ. Keywords: Padovan sequence, finite fields, matrix order, lifting modulo prime powers

Zenodo (CERN European Organization for Nuclear Research)
Alma Mater Europaea (SI)
Openalex Percentile: Top 5%
Coding theory and cryptography
DOIOpen Access PDFOpenAlex
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"On Matrix Order Growth Modulo Prime Powers: The Padovan Case" — Pietro Franesi · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS