This research determines matrix order in GL₃(ℤ/9ℤ), suggesting growth patterns under prime power lifting.
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.
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Pietro Franesi (2026) studied this question.
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