Topological Representation and Adaptation of Special Integer Systems within Fractal Number Theory (FNT)

Author: Zhang Suhang, Luoyang, Henan

The topological recursive representation system of Fractal Number Theory (FNT) is not limited to primitive primes. All common integer types and special number‑theoretic objects, including composite numbers, special‑form primes, pseudoprimes and derived integer sequences, can be incorporated into the FNT framework for topological modelling and structural analysis, forming a fractal representation system covering the full domain of integers.

This chapter strictly maintains the boundaries of the paradigm: the adaptation of FNT to various categories of integers introduces new structural perspectives, novel topological characterizations and additional evolutionary interpretations. It does not replace the algebraic criteria, modular arithmetic, analytic estimation and primality‑testing methods of traditional number theory, and always exists in complementary coexistence with the classical system.

I. Topological Representation of Composite Numbers via FNT

Classical number theory treats composite numbers through multiplicative decomposition, regarding them as product combinations of prime factors and focusing on static numerical decomposition relations.

Within the FNT framework, a composite number may be viewed as a fractal topological body generated by superposition, composition and nesting of multiple fractional‑scale operator groups.

Pan‑factorization can be applied to strip away its scaled nested structure layer by layer, observing the hierarchical distribution, self‑similar textures and folding features formed after the superposition of operators at distinct prime‑derived scales.

Compared with the simple product decomposition in classical theory, FNT is capable of revealing the generative origin and hierarchical constitution of composite numbers at the structural level.

II. Scope of FNT Adaptation for Special‑Form Primes

All primes defined by special construction rules can be individually modelled with dedicated fractal‑evolution representations, including but not limited to:

- Mersenne primes

- Twin primes

- Fibonacci primes

- Factorial primes

- Palindromic primes

- Sophie Germain primes

In the classical system, such primes are mainly defined by algebraic constructions, recurrence relations and congruence conditions.

Under FNT, according to their specific construction rules, these primes can be mapped onto fractal topological manifolds with designated iteration rhythms, scale constraints and recursive boundaries.

Different categories of special‑form primes exhibit differentiated fractal curvature, nesting periods and evolutionary density characteristics, enabling structural‑level distinction between ordinary primes and constructible primes.

III. Structural Identification of Pseudoprimes and Anomalous Composite Numbers by FNT

The family of pseudoprimes constitutes a special class of “prime‑like structures” in number theory:

- Fermat pseudoprimes

- Euler pseudoprimes

- Strong pseudoprimes

- Carmichael numbers (absolute pseudoprimes)

Classical number theory can only judge that their numerical behaviour resembles primes via modular operations, yet cannot intuitively explain the structural root cause of why a composite number may simulate the congruence properties of primes.

FNT offers a new interpretive path from the topological viewpoint:

Although pseudoprimes satisfy primality‑related congruence conditions numerically, identifiable topological differences exist between their fractal recursive structures, scale‑nesting patterns, operator‑evolution maps and those of genuine primes.

Through FNT modelling, one may visually distinguish numerical behavioural resemblance from essential topological disparity, providing a structural basis for the discrimination and genesis of pseudoprimes.

IV. Unified Logic for Global Adaptation

Whether for primes, composite numbers, special‑form primes or pseudoprimes, all integers possess underlying features of recursiveness, decomposability and scalability.

Consequently, every class of integers may be trialled for integration into the FNT framework:

- Classical number theory: investigates numerical, algebraic and statistical properties of numbers

- FNT number theory: studies generative structures, evolutionary topology, scale hierarchies and self‑similar features of numbers

The two systems operate in parallel and complement each other across the whole domain.

V. Summary of Paradigm Boundaries

1. FNT enables topological modelling and structural characterisation for all classes of integers; its adaptive scope covers conventional integers and special number‑theoretic sequences.

2. FNT does not displace classical methods including traditional primality testing, congruence analysis, algebraic construction and analytic statistics.

3. The value of FNT lies in complementing the dynamical‑structural dimension of numbers, extending number theory from the study of static numerical properties to dynamical generation and topological evolution.

With this development, Fractal Number Theory completes its paradigmatic expansion from the study of individual primes toward compatibility with a full‑domain integer system.

VI. FNT Fills the Long‑Standing Gap of Visualizable Topology in Number Theory

Whether algebraic or analytic, traditional number theory relies centrally on symbolic formulae, algebraic derivation and analytic estimation for its research, forming a generally static, non‑visual theoretical system.

Classical analytic number theory only produces numerical function plots and discrete coordinate diagrams for zeros; it lacks representations of the endogenous growth, recursive nesting and structural evolution of integers themselves. Algebraic number theory is built entirely upon abstract symbolic operations, with no native geometric visualisation system. For a long time, number theory has only described numerical values, formulae and statistical laws, while omitting the ontological structure and dynamical topological morphology of numbers.

Fractal Number Theory (FNT) remedies this core deficiency: relying on the iteration of fractional‑scale operators and the recursive structure of FCFG, all integers and number‑theoretic objects are endowed with dynamically evolvable, intuitively visualisable topological ontologies.

This establishes the complete three‑layer foundation of number theory:

1. Algebraic foundation: defines operational rules for numbers

2. Analytic foundation: describes statistical fluctuations of numbers

3. FNT topological foundation: presents structural generation and dynamical evolution of numbers

FNT supplies the whole number‑theoretic system with the visualisation and structural‑topology dimension that has been absent for centuries.