The classical theory of numbers rests on well-established foundations, notably the distinction between prime and composite numbers, as well as the notion of divisibility. However, these definitions, though functional, present conceptual limitations when applied to the extremities of the set of whole numbers, particularly in the cases of zero and one. These two numbers, often treated as exceptions or marginal cases, reveal implicit inconsistencies in the way divisibility is traditionally formulated. In this study, we propose a radical reformulation of the structure of whole numbers through the introduction of two fundamental concepts: the ultimate number and the ultimate divisor. These notions are based on an innovative approach to divisibility, which integrates the relative value of the divisors that is, their position below or above the number in question. This perspective allows for the redefinition of the properties of all whole numbers, including those that elude classical classifications.

Grounded in a novel mathematical framework, this study partitions the set of whole numbers (ℕ0) into four distinct hierarchical classes. A key innovation is the definition of Ultimate Numbers—the union of the prime numbers with zero and one—which resolves classic conceptual limitations. Three further subsets, representing increasing degrees of numerical complexity, are subsequently defined by the initial distinction between ultimate and non-ultimate numbers within ℕ0. The structural interaction among these four classes yields unique arithmetic arrangements in their initial distribution, most notably revealing an exact and recurring 3:2 ratio.