Part Two: Axioms and Mathematical Formulas

—— Formal Foundations of the Composite Number Construction System

Author: Zhang Suhang (Luoyang, Henan)

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Abstract

Based on the analogical framework of “primes–composites” and “fractal primitives–fractal patterns” established in Part One, this paper extracts three basic axioms of the construction system and presents their corresponding mathematical expressions. The focus is on:

1. Formally distinguishing between “regular composites” and “disordered hybrid composites”;

2. Clarifying the scope and constraints of each formula;

3. Re-emphasizing that the terms “disordered” and “chaotic” in this paper refer only to combinatorial diversity of factor composition and non-recursive numerical distribution, not to dynamical systems chaos.

This paper provides directly computable mathematical tools for Part Three: “Fractal-Coefficient Coupled Evolution”.

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1. Basic Axioms

Axiom 1 (Construction Primitive Axiom)

Prime numbers are independent, indivisible construction primitives in the natural numbers.

Formalization: Let \mathbb{P} = \{2,3,5,7,11,\dots\} be the set of all primes. Any integer n>1 can be expressed as a product of primes, and if the order of factors is constrained, the representation is unique.

Axiom 2 (Unique Factorization Axiom)

Any composite number N (N>1, not prime) can be uniquely decomposed into a product of prime powers:

N = \prod_{i=1}^{k} p_i^{a_i}, \quad p_i \in \mathbb{P},\ a_i \in \mathbb{N}^+,\ k \ge 2 \ (\text{or } k=1 \text{ but } a_1\ge 2)

\]  

where p_1 < p_2 < \dots < p_k. This is the standard form of the Fundamental Theorem of Arithmetic.

Axiom 3 (Composition Structure Axiom)

The structural properties of a composite number (including its magnitude, number of factors, and distribution regularity) are completely determined by the set of prime factors \{p_i\} and their exponents \{a_i\}.

Corollary: Changing the choice of primes or their exponents changes the “construction parameters” of the composite, analogous to changing the scaling factor, iteration count, or offset in fractals.

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2. Formal Definition of Composite Classification

To align with the “order → chaos” analogy in fractals, composites are explicitly divided into two categories:

2.1 Regular Composites (Ordered Composites)

Definition: There exists a fixed finite set of primes \mathcal{P}_0 = \{q_1, q_2, \dots, q_m\} (m \ge 1) and exponent range constraints (e.g., an upper bound on exponents or exponents satisfying a linear recurrence) such that all prime factors of the composite belong to \mathcal{P}_0.

Typical examples:

· Powers of a single prime: 2^a

· Numbers generated by two primes: 2^a 3^b (a,b \ge 0, at least one positive) — i.e., 5-smooth numbers.

· Arbitrary power combinations of a fixed prime set: \prod_{i=1}^m q_i^{a_i}, where each a_i can be any nonnegative integer (not all zero).

Property: The distribution of such numbers in the natural numbers follows describable regularities (e.g., the sequence 2^a 3^b can be generated recursively, and adjacent ratios tend to \log 2 / \log 3, etc.), analogous to regular patterns in fractals generated by fixed iteration parameters.

2.2 Disordered Hybrid Composites (Chaotic Composites)

Definition: The set of prime factors of the composite cannot be contained in any fixed finite set of primes (i.e., as the number grows, new primes continually appear), or its exponent combinations lack simple global recurrence constraints.

A more operational definition:

Let \omega(N) be the number of distinct primes in the prime factorization of N. If as N \to \infty, \omega(N) is unbounded (can become arbitrarily large) and the appearance of prime factors follows no fixed pattern, then N is called a disordered hybrid composite.

Typical examples:

· Product of the first k primes: 2 \times 3 \times 5 \times 7 \times \dots \times p_k (primorials).

· Random selection of several distinct primes (e.g., 2 \times 7 \times 13 \times 23) and mixtures of powers, where the prime factor set is not confined to a small fixed set.

Property: The numerical distribution intervals are irregular and cannot be described by a finite-parameter recurrence formula for all such composites.

Important note: The terms “disordered” and “chaotic” here do not refer to dynamical systems chaos, but rather to:

(1) Combinatorial explosion of prime factor sets (infinitely many possibilities);

(2) The positions on the number line lack simple self-similar recursive laws.

This usage is consistent with the everyday sense in fractal geometry of “parameter randomization leading to visual disorder,” not a strict mathematical definition.

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3. Standard Formulas

3.1 General Prime Factorization Formula (already given)

N = \prod_{i=1}^{k} p_i^{a_i}, \quad p_i \in \mathbb{P},\ a_i \ge 1,\ k \ge 1,\ p_1 < p_2 < \dots < p_k

\]  

When k=1,\ a_1=1, N is prime; when k\ge 2 or k=1,\ a_1\ge 2, N is composite.

3.2 Hierarchical Combination Recurrence

Define a family of composite sets \mathcal{C}_r (r \ge 2 is the total number of prime factors counted with multiplicity):

\mathcal{C}_r = \left\{ N = \prod_{i=1}^{k} p_i^{a_i} \ \middle|\ \sum_{i=1}^{k} a_i = r,\ a_i\ge 1,\ k\ge 1 \right\}

\]  

Then:

· \mathcal{C}_2: products of two prime factors (may be equal), e.g., p^2,\ pq.

· \mathcal{C}_3: products of three prime factors (with multiplicity), e.g., p^3,\ p^2q,\ pqr.

Significance of this recurrence: Layering by the total number of prime factors (with multiplicity) is analogous to layering by “iteration count” in fractals. Regular composites are typically confined to subsets \mathcal{C}_r within a fixed prime set, whereas disordered composites continually introduce new primes as r increases.

3.3 Power Construction Expression (for Regular Composites)

Given a fixed prime set \mathcal{P}_0 = \{q_1,\dots,q_m\}, the set of all composites (including 1, if we agree a_i=0 corresponds to the factor 1) generated by them is:

S(\mathcal{P}_0) = \left\{ \prod_{i=1}^{m} q_i^{a_i} \ \middle|\ a_i \in \mathbb{N} \right\}

\]  

This set is closed under multiplication, and the composites in its complement are exactly the disordered hybrid composites relative to \mathcal{P}_0 (those containing at least one prime factor not in \mathcal{P}_0).

In particular: if \mathcal{P}_0 = \{2,3\}, then S(\{2,3\}) = \{2^a 3^b\} is the set of all 5-smooth numbers. Their distribution density in the natural numbers is asymptotically \frac{1}{2\log 2 \log 3} \cdot \frac{(\log N)^2}{N} (Dirichlet hyperbolic summation), exhibiting clear analytic regularity.

3.4 Disorder Measure (optional, as preparation for Part Three)

One may define a “prime factor entropy” concept (not probabilistic, merely heuristic):

H(N) = \text{number of distinct prime factors} = \omega(N)
\]

or more finely:

H_{\text{weight}}(N) = \sum_{i=1}^{k} \frac{1}{a_i+1}
\]

When H(N) grows without bound and without regularity as N increases, it can be regarded as a quantitative indicator of “disorder.” This indicator involves no dynamics; it is merely a combinatorial statistic.

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4. Correspondence Table of Axioms and Fractal Parameters (Solidifying the Analogy)

Fractal Geometry Concept Number Theory Formula/Set Explanation
Initial figure Single prime p Indivisible primitive
Iteration rules (scaling, offset) Exponent a_i and prime choice p_i Determine how to “combine”
Iteration count Total prime factor multiplicity r = \sum a_i Corresponds to hierarchical depth
Regular fractal pattern Regular composite set S(\mathcal{P}_0) Fixed prime set, arbitrary exponents
Parameter randomization / disorder Disordered hybrid composites (prime factor set has no fixed finite bound) Combinatorial explosion, no global recurrence

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5. Conclusion and Preview of Next Part

This paper formalizes the construction process “primes → composites” at the axiomatic level, strictly distinguishes regular composites from disordered hybrid composites, and clarifies the non-dynamical-system semantics of the term “chaos” in this series. The provided formulas (hierarchical combination recurrence, power construction, disorder measures) offer operational tools for numerical experiments and graphical mapping.

Part Three will introduce fractal-coefficient coupled evolution:

· Fractal side: scaling factor s, iteration coefficient \lambda, offset coefficient \delta;
· Number theory side: let s correspond to prime magnitude, \lambda to exponent growth rate, \delta to the rule for expanding the prime set.
By adjusting these “numerical parameters,” we will demonstrate a continuous evolutionary spectrum from primes to regular composites to disordered hybrid composites, and provide side-by-side comparisons of numerical distribution graphs and fractal patterns.

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References (illustrative)

[1] Fundamental Theorem of Arithmetic, any elementary number theory textbook.
[2] Distribution of smooth numbers, de Bruijn (1951).
[3] Parameter space in fractal geometry, Mandelbrot (1982).
[4] Author’s series on “number theory–fractal analogy” (2026).

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(End of Part Two. All formulas are verifiable within elementary number theory; no dynamical system equations are involved.)