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arXiv · 2608.30069

Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices

Wilson Arley Martinez·Samin Ingrith Ceron
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Abstract

We study matrix representations of the Jacobsthal sequence generated by binary 3x3 matrices with determinants 0, 2, and -2, using linear algebraic methods analogous to Fibonacci-type constructions. Explicit formulas for matrix powers are obtained, yielding identities for Jacobsthal numbers, including convolution formulas, trace relations, determinant expressions, and Cassini-type identities. We further derive congruence relations and recurrence formulas, and analyze arithmetic properties such as partial sums and the Sidon-type structure of the sequence. Finally, we prove that exactly three conjugacy classes of binary 3x3 matrices generate the Jacobsthal sequence, providing a unified algebraic framework that connects matrix theory with second-order linear recurrences.

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Wilson Arley Martinez, Samin Ingrith Ceron. 2026-08-30. Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices. https://arxiv.org/abs/2608.30069

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