"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