The Novel Concept of Ultimate Numbers

Grounded in a novel mathematical framework, this study partitions the set of whole numbers (ℕ0) into only two distinct whole number 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. Thus the set ℕ0 is considered to consist of two primary sets of whole numbers at absolute properties. In various closed matrices, the starting geometric distribution of sequences of these two whole number sets is organized into exact 3:2 or 1:1 ratios. Extension of Sophie Germain number concept to these two primary sets also generates similar and singular arithmetic phenomena in different initial closed matrices.