This article analyzes the periodic orbits of Syracuse dynamical systems in a novel algebraic setting: the commutative ring of graded $n$-adic integers. Within this context, this article introduces a dual-radix modular division algorithm for computing the graded canonical expansions and graded quotients for a certain class of rational expressions that arise from periodic orbits within these dynamical systems. This division algorithm yields two novel methods for testing the integrality of the B\"{ o}hm-Sontacchi numbers.
Categories: math.NT