Fractal Number Theory (FNT) — An Original Number‑Theoretic Branch Built upon the FCFG Framework

Author: Suhang Zhang
Affiliation: Luoyang, Henan

Abstract

Based on the recursive architecture of Fractal Continued‑Fraction Geometry (FCFG), this paper establishes a novel, self‑consistent and rigorously derivable system of Fractal Number Theory (FNT). Supported by three cornerstones — hierarchical structure, embedding of fractal dimension and continued‑fraction recursive mapping — the system breaks through the paradigms of classical number theory, Euclidean geometry, Riemannian topology and conventional lattice systems. Through quantitative characterization of scaling dimension, binding of hierarchical recursion and unification of topological structures, a deep fusion of number theory, fractal dimension analysis and geometric topology is realized.

This paper formulates three criteria for identifying a new mathematical branch, and constructs a bijective recursive correspondence: hierarchy ↔ continued‑fraction term ↔ fractal scaling. The resulting framework supplies a brand‑new underlying deductive tool for problems concerning algebraic numbers, transcendental numbers, dynamical systems and scaling topology, constituting an independent and complete original number‑theoretic system.

Keywords: Fractal Number Theory (FNT); Fractal Continued‑Fraction Geometry (FCFG); continued‑fraction recursion; fractal dimension; hierarchical topology

1. Introduction

Classical number theory centers on integers, divisibility, congruences, prime distribution and algebraic number fields. Its geometric foundation rests on Euclidean uniform flat space, characterized by constant scale, discrete structure and the absence of hierarchical recursion. Traditional fractal geometry only describes scaling and dimensional features of figures without involving number‑theoretic recursion; the classical theory of continued fractions merely serves as an approximation tool for real numbers and lacks hierarchical topology as well as dimension embedding.

Number theory, fractal geometry and topology have long been isolated from one another. They cannot explain the deep interrelations among intrinsic properties of numbers, iterative structures, spatial scales and topological continuity.

Accordingly, upon the native recursive framework of FCFG, this paper reconstructs the underlying logic of number theory. By embedding fractal dimensions into the hierarchical system and taking continued‑fraction recursion as the global core mapping, Fractal Number Theory (FNT) is constructed as an original mathematical branch distinct from all classical systems.

2. Definition of FNT and Criteria for a New Mathematical Branch

2.1 General Definition

Fractal Number Theory (FNT) is an original number‑theoretic branch built on FCFG. It adopts hierarchical topological structure as the spatial carrier, embedding of fractal dimension as the quantitative rule, and continued‑fraction recursive mapping as the core deduction mechanism.

The system is supported by three tightly coupled core components:

1. Hierarchical topological structure (fundamental spatial carrier)
2. Embedding formula of fractal dimension (core scaling operation)
3. Continued‑fraction recursive mapping (global iterative logic)

These three layers are deeply integrated to form a self‑consistent deductive system for number theory.

2.2 Three Criteria for an Independent Mathematical Branch

To qualify as a genuinely new mathematical branch, a theory must simultaneously satisfy three sufficient‑and‑necessary conditions: independent research object, independent core operation, independent recursive structure. FNT fully meets these requirements.

(1) Independent Underlying Research Object

FNT takes the hierarchical recursive topological system derived from FCFG as its exclusive research substrate. This novel mathematical space permits unlimited iteration and scaling, possesses ordered hierarchies and self‑similar recursion. It is entirely different from Euclidean flat space, Riemannian curved manifolds and conventional discrete lattices, with its own research scope and theoretical boundary.

(2) Independent Core Operation

The standard fractal‑dimension formula acts as the unique core tool for scaling deduction:
D=\frac{\ln m}{\ln(1/r)}
Dimension parameters are systematically embedded into every hierarchy so that hierarchical topology, continued‑fraction iteration and scale transformation can be quantified precisely. This operation is not a conventional algebraic operation, differential operation or ordinary topological operation; it is a dimension‑scaling calculus exclusive to FNT.

(3) Independent Recursive Logical Structure

FNT establishes a global bijective mapping:
hierarchical sequence ↔ continued‑fraction iterative term ↔ fractal scaling factor

Iterative evolution, convergence, discreteness and continuity of the whole system are governed by continued‑fraction recursion. This native recursive structure does not exist in classical number theory, traditional fractal geometry or standard topology and represents an original recursive logic.

In summary, FNT satisfies all three conditions and qualifies as a brand‑new original branch of mathematics.

3. Core Mechanism: Unified Mapping among Hierarchy, Dimension and Continued Fraction

3.1 Mathematical Implication of Hierarchical Structure

The FCFG framework naturally features layered iteration. Each hierarchy corresponds to a set of self‑similar topological units. Greater hierarchical depth yields finer iteration and subdivision, and the structure gradually approximates a continuous manifold.

Hierarchical structure determines the discreteness, fineness and topological connectivity of the system. It is the fundamental carrier that geometrizes and scales number‑theoretic properties.

3.2 Global Embedding of Fractal Dimension

Classical number theory is formulated within spaces of fixed dimension and lacks a mechanism for dynamic variation of scaling dimension.

In FNT, fractal dimension runs through the entire deduction:

- Low‑dimensional hierarchies correspond to discrete number‑theoretic structures
- High‑dimensional hierarchies correspond to continuous topological manifolds
- Dynamic variation of dimension directly controls approximation precision, convergence behaviour and topological morphology

Thereby a dynamic number‑theoretic system with controllable dimension and variable scale is constructed.

3.3 The Governing Role of Continued‑Fraction Recursion

Periodicity and non‑periodicity of continued fractions inherently match the distinction between algebraic numbers and transcendental numbers:

- Periodic continued fractions → algebraic numbers, finite hierarchies, discrete fractal structures
- Non‑periodic continued fractions → transcendental numbers, infinite hierarchies, continuous manifold structures

Via the core recursive mapping, FNT realizes four‑dimensional unification of number‑theoretic attribute, hierarchical depth, fractal dimension and topological morphology, and connects the underlying logics of multiple disciplines.

4. Innovation and Theoretical Value of the System

1. Fundamental innovation: Abandoning the inherent frameworks of classical geometry and traditional number theory to establish a brand‑new underlying mathematical logic.
2. Structural innovation: Building a dynamic hierarchical number‑theoretic system governed globally by continued‑fraction recursion.
3. Dimensional innovation: Introducing fractal dimension formally into number theory so that attributes of numbers can be deduced together with spatial scaling.
4. Integrative innovation: Unifying number theory, fractal geometry and topology to form a cross‑disciplinary complete original mathematical system.

The proposed system provides new theoretical reconstruction tools and deductive routes for classical problems including prime distribution, zero‑point problems, number‑field topology, construction of transcendental numbers and iterations in dynamical systems.

5. Conclusion

Based on the foundational framework of FCFG, this paper constructs the complete theoretical system of Fractal Number Theory (FNT). FNT possesses an independent research object, independent core operation and independent recursive topological structure, fully satisfying the criteria for founding a new mathematical branch.

Through deep coupling of hierarchical topology, embedding of fractal dimension and continued‑fraction recursive mapping, FNT achieves unified modelling of number theory, geometric scaling and topological iteration. It forms a self‑consistent, expandable original mathematical system and supplies a new theoretical substrate for reconstruction, analysis and deduction of core problems in number theory.