Part One: Analogical Foundation in Number Theory

—— The Fractal Generation Logic of Composite Numbers

Author: Zhang Suhang, Luoyang, Henan

Abstract

Starting from the constructive processes of number theory and fractal geometry, this paper proposes an analogical framework: prime numbers are like the minimal initial figures in fractals, while composite numbers are generated through the combination and iteration of primes—a process homologous to the evolutionary logic in fractal geometry of “simple figure → repeated iteration → hierarchical superposition → complex morphology.” This analogy aims to provide an intuitive foundation for subsequently establishing the evolutionary chain from “primes → ordered composites → disordered composites.” It should be noted that the terms “disordered” and “chaotic” in this paper refer only to the diversity of factor composition and irregularity of numerical distribution, not to chaos in the sense of dynamical systems.

1. Basic Understanding

1.1 Prime Numbers: Indivisible Construction Primitives

In the natural number system, a prime is defined as an integer greater than 1 that has no positive divisors other than 1 and itself. From a constructive perspective, primes are “indivisible” minimal units—any composite number can be uniquely decomposed into a product of primes (the Fundamental Theorem of Arithmetic). This resembles the initial figures used for iteration in fractal geometry (e.g., a line segment, an equilateral triangle), which themselves are not composed of smaller similar figures.

1.2 Composite Numbers: Products of Primes

A composite number is obtained by multiplying two or more primes, with the same prime allowed to repeat (exponential form). For example:

· 6 = 2 × 3 (combination of two distinct primes)

· 8 = 2³ (power of a single prime)

· 12 = 2² × 3 (mixed powers)

This process is structurally isomorphic to the fractal construction operation of “scaling, rotating, translating, and repeatedly joining initial figures”: the primitives remain unchanged, while the combination rules determine the final form.

2. Core Analogy

Fractal Geometry Number Theory (Composite Construction)

Simple base figure (e.g., line segment, triangle) Prime numbers (indivisible primitives)

Iteration rules (scaling, rotation, translation coefficients) Choice of primes, exponents, order of combination

Hierarchical superposition (multiple rule applications) Repeated multiplication of prime factors (including powers)

Complex geometric shapes (e.g., Koch snowflake, Sierpinski carpet) Diverse composites (from 6 to very large composites)

Key analogical points:

· In fractals, merely changing iteration parameters (e.g., scaling factor, offset) can produce vastly different shapes from the same initial figure (from regular fractals to nearly random forms).

· In number theory, merely changing the set and exponents of prime factors can yield radically different composites from the same prime primitives:

  · Regular composites: e.g., 2^a·3^b, 2^a·5^b – their numerical distribution shows clear regularity (generated by a fixed set of primes).

  · Disordered hybrid composites: e.g., 2×3×5×7×11×13, or random mixtures of many distinct primes – their factor structure is complex and their numerical distribution lacks simple recurrence relations.

3. Core Thesis

The evolutionary law of the numerical system and the growth law of fractal patterns share a homologous similarity. Specifically:

· From simple to complex: single prime → product of two primes → mixture of multiple prime powers → random combination of many distinct primes.

· From ordered to structurally diverse: regular composites generated by a fixed set of primes (like regular patterns in fractals with fixed parameters) → disordered composites with unrestricted combinations (like fractal patterns that lose visual regularity when parameters are randomized).

3.1 Special Note on the Term “Chaos”

The terms “disordered” and “chaotic” appearing in this paper and subsequent parts refer only to:

1. Combinatorial explosion of factor composition, which cannot be described by a finite fixed set of primes;

2. The distribution intervals of composites on the number line lacking simple recursive laws.

This is not chaos in the dynamical systems sense (i.e., long-term unpredictability due to sensitive dependence on initial conditions in deterministic systems). Even in fractal geometry, certain parameter choices can lead to “chaotic behavior” (e.g., the bifurcation diagram of the logistic map), but the generation of composites in number theory is a discrete combinatorial process without iterative dynamics. Hence, the analogy in this paper is about structural similarity in construction logic, not mathematical equivalence. This distinction is a prerequisite for the entire argument.

4. Conclusion and Preview of Next Part

This paper establishes a structural analogy between “primes–composites” and “fractal primitives–fractal patterns,” clarifies the distinction between “regular composites” and “disordered hybrid composites,” and cautions against cross-contextual use of the term “chaos.” Based on this, the second part will extract the axioms and mathematical formulas of this constructive system, solidifying the analogical relations into operational mathematical expressions (general prime factorization formula, hierarchical combination recurrence, power construction expression), laying a rigorous foundation for the “fractal-coefficient coupled evolution” in the third part.

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(Note: This is the first paper in a series, focusing on establishing the analogy. All mathematical details will be presented in the second part. Readers interested in a precise definition of “regular composites” may refer to the exponent constraints given in the second part.)