According to a new mathematical definition, whole numbers are divided into two sets, one of which is the merger of the sequence of prime numbers and numbers zero and one. Three other definitions, deduced from this first, subdivide the set of whole numbers into four classes of numbers with own and unique arithmetic properties. The geometric distribution of these different types of whole numbers, in various closed matrices, is organized into exact value ratios to 3/2 or 1/1.
The ultimate numbers and the 3/2 ratio
The ultimate numbers
Definition of an ultimate number
Considering the set of whole numbers, these are organized into two sets: ultimate numbers and non-ultimate numbers.
Ultimate numbers definition:
An ultimate number not admits any non-trivial divisor (whole number) being less than it.
Non-ultimate numbers definition:
A non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Other definitions
Let n be a whole number (belonging to ℕ), this one is ultimate if no divisor (whole number) lower than its value and other than 1 divides it.
Let n be a natural whole number (belonging to ℕ), this one is non-ultimate if at least one divisor (whole number) lower than its value and other than 1 divides it.
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The progressive differentiation of source classes and final classes of whole numbers is organized into a powerful arithmetic arrangement generating transcendent ratios of value 3/2. Thus, the source set of whole numbers includes, among its first ten numbers, 6 ultimate numbers against 4 non-ultimate numbers. The next source set, that of the non-ultimates, includes, among its first ten numbers, 4 raised numbers against 6 composite numbers. Finally, the source set of composites includes, among its first ten numbers, 6 pure composites against 4 mixed composites.
A very strong entanglement links all these sets of numbers which oppose in multiple ways in ratios of value 3/2 (or reversibly of ratios 2/3). For example, the first 6 ultimates (0-1-2-3-5-7) are simultaneously opposed to the 4 non-ultimates (4-6-8-9) among the first 10 natural numbers, to the 4 raiseds of the first 10 non-ultimates (4-8-9-16) and to the 4 ultimates beyond the first 10 whole numbers (11-13-17-19).
This entangled classification of whole numbers makes it possible to define a set of forty primordial numbers. These forty primordial numbers are the set of first ten numbers in each of the four final classes of whole numbers. It is understood that the term "primordials" designates these forty primordial numbers.
Within this table operate multiple singular arithmetic phenomena depending on the ultimate or non-ultimate nature of the values of these hundred sums and their geographic distribution including mainly various 3/2 value ratios often transcendent.