Generalized Factorization and Fractal Number Theory（FNT）

Author: Zhang Suhang

Luoyang School of Mathematics

Abstract

Traditional number theory takes the Fundamental Theorem of Arithmetic in the ring of integers as its absolute cornerstone, regarding primes as the unique, indivisible atoms in the multiplicative structure. Based on the theory of prime fractions in FCFG (Fractal Continued Fraction Geometry), this paper proposes the concept of “Generalized Factorization,” elevating the semantics of “factorization” from a discrete multiplicative structure to the generation and deconstruction of fractal nested topology. This paper explores how this paradigm shift redefines the geometric identity of primes, how continued fraction recursion can be extended into spatial evolutionary dynamics, and introduces fractal curvature to characterize the geometric features of recursive structures. It points out that this provides a purely topological-fractal path that bypasses arithmetic exhaustive search for computational number theory (the hard problem of large integer factorization) and post-quantum cryptography. This study aims to open a new branch for the deep integration of number theory and fractal geometry—Fractal Number Theory—and proposes the core axiomatic framework that this branch urgently needs to address.

Keywords: Generalized Factorization; Fractal Number Theory; prime fractions; prime factors; Fundamental Theorem of Arithmetic; continued fraction recursion; fractal curvature

I. Introduction: The Boundaries of the Fundamental Theorem of Arithmetic and Fractal Dimensional Elevation

For two thousand years, the foundation of number theory has been deeply rooted in the arithmetic operations of integers. The Fundamental Theorem of Arithmetic (any positive integer greater than 1 can be uniquely decomposed into a product of primes) establishes the absolute central position of primes in the discrete multiplicative world. Traditional number theory research, whether Diophantine equations, analytic number theory, or algebraic number theory, essentially revolves around “addition, subtraction, multiplication, division” and “the distribution of primes.”

However, as mathematics enters the era of high dimensions and complex systems, purely arithmetic deduction has gradually shown its fatigue in the face of large integer factorization (such as the RSA cryptosystem) and multidimensional data complexity. The mathematical community has long overlooked an underlying perspective: the essence of “factorization” may not be limited to finding “smaller integer multipliers,” but rather to finding “more fundamental generative structures.”

Based on the FCFG system (Fractal Continued Fraction Geometry), this paper re-examines the Fundamental Theorem of Arithmetic and introduces the concept of prime fractions (1/p), proposing Generalized Factorization in an attempt to lead number theory from the discrete kingdom of arithmetic into the vast territory of fractal geometry.

II. Generalized Factorization: From Arithmetic Splitting to Fractal Deconstruction

2.1 Comparison between Traditional Factorization and FCFG Factorization

· Old perspective (arithmetic splitting): Factorization = splitting an integer into a product of several primes (e.g., 15 = 3 × 5). This is a parallel, static, discrete additive and multiplicative structure.

· New perspective (fractal deconstruction): Factorization = deconstructing a complex geometric body with self-similar and nested structures into a sequence of topological generating operators composed of prime fractions (1/p).

2.2 Mathematical Connotation of Generalized Factorization

In the FCFG framework, the prime fraction 1/p is not merely the reciprocal of a prime p; it is a native fractal basis with the capacity for “infinite nested folding.” Generalized Factorization does not negate the Fundamental Theorem of Arithmetic; rather, it elevates it: it studies how a composite fractal manifold can be deconstructed into a combination of prime fraction operators. This expands the research object of number theory from pure “numbers” to “recursive structures of scale scaling.”

Each prime fraction basis corresponds to a set of recursive folding rules, and these rules determine the characteristic fractal curvature of the fractal structure at the corresponding level. Larger primes correspond to smaller prime fraction values, with finer inward folding, and can generate a set of highly complex curvature features; smaller primes correspond to relatively looser folding and possess another curvature spectrum. Curvature becomes a geometric invariant characterizing the local bending morphology of fractal space.

III. Geometrization of the Prime Itself

In the traditional system, a prime p is a static “building block” used to construct larger integers. But in Generalized Factorization, the identity of primes undergoes a fundamental change:

Through the prime fraction 1/p, a prime p becomes a basis operator defining spatial stretching, topological folding, and fractal curvature.

This means that a prime not only represents an isolated numerical value; it also defines:

1. A specific infinite recursive structure of continued fractions;

2. A scale-scaling rule for fractal space;

3. A frequency band carrying microscopic differential information;

4. The fractal curvature features at the corresponding level.

This brings about entirely new research questions:

· Will the macroscopic distribution of primes have a deep intrinsic connection with the morphology of the curvature spectrum of fractal space?

· Can the geometric evolutionary properties of FCFG be used to observe, or even derive, the patterns of prime sequences in reverse?

This provides a visualizable, geometric language for prime research, freeing it from the limitations of purely algebraic and analytic derivation.

IV. A New Bridge Connecting Number Theory and Geometry

Although algebraic geometry has already established connections between number theory and geometry (such as algebraic curves, schemes, and field extensions), its core path remains algebraic. FCFG provides a completely new path relying on “continued fractions + fractal recursion.”

Continued fractions have long been deeply tied to number theory (such as quadratic irrationals and Diophantine approximation). FCFG amplifies the arithmetic recursion of continued fractions into a complete geometric manifold. This means that a large number of classical number theory problems can be equivalently transformed into problems of topological evolution and curvature evolution on fractal spaces. Highly abstract objects of algebraic number theory thus acquire measurable and visualizable geometric forms.

V. The Turn in Computational Number Theory: Beyond Arithmetic Exhaustive Search

In computational number theory, large integer factorization is a century-old hard problem. Existing algorithms (such as the quadratic sieve and the number field sieve), and even the quantum Shor algorithm, are essentially still searching for factors at the level of “arithmetic and periods.”

Generalized Factorization proposes a completely different direction: encoding numbers into fractal topological structures and using topological-fractal deconstruction paths for processing.

· We do not need to stubbornly persist within the traditional integer arithmetic domain.
· We can explore computational paradigms based on prime fraction topological transformations and fractal curvature features.
· This provides a foundational mathematical-physical approach for post-quantum security that differs from “lattice-based cryptography” or “hashing.”

VI. Paradigm Shock: The Research Objects of Fractal Number Theory

With the addition of Generalized Factorization, the research goals of number theory are greatly expanded. Classical number theory answers “into which primes can this number be factored”; Generalized Factorization further asks “by which prime fraction bases is this structure recursively generated, and what curvature features does it possess.”

Thus, we formally open the branch “Fractal Number Theory,” whose new research objects include:

1. The fractal dynamics of prime fraction operators;
2. The algebraic representation of self-similar nested structures;
3. Measure, dimension, and fractal curvature theory on continued fraction recursive spaces.

VII. Discussion and Prospects: Core Theorems Urgently to Be Solved

It must be objectively acknowledged that Generalized Factorization is currently still at the stage of paradigm construction. For it to become a rigorously accepted mathematical system, the following rigorous mathematical constructions must be completed in the future:

1. Strictly define the operation rules on the space of prime fraction operators;
2. Establish a formal definition of fractal curvature adapted to this recursive space;
3. Prove the “existence” and “uniqueness” theorems corresponding to Generalized Factorization;
4. Establish a rigorous mapping relationship between Generalized Factorization and the classical Fundamental Theorem of Arithmetic;
5. Derive new number-theoretic propositions with computability.

VIII. Conclusion

Generalized Factorization extends the research territory of number theory from pure integer arithmetic outward into the field of fractal continued fraction geometry. Primes acquire new identities as fractal generators and curvature regulators, and the core concept of “factorization” is reinterpreted. This is not only an innovation of mathematical tools but also a leap in cognitive paradigm, opening an independent and profound intellectual path for computational number theory, algebraic geometry, and cryptography.