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The Significance of Prime Fractions for Complex Numbers

Author: Zhang Suhang

School: Luoyang School of Mathematics

Abstract

Traditional algebraic number theory, when dealing with the complex domain, has to restrict the complex basis (a,b) strictly to a discrete integer ring—the “Gaussian integers”—in order to preserve the “uniqueness theorem” of factorization. Although this compromise preserves the arithmetic system, it sacrifices the continuity and topological evolutionary characteristics of the complex plane. Based on the FCFG fractal continued-fraction geometry system, this paper explores the fundamental reconstruction of the complex domain and the complex plane brought about by “prime fractions (1/p).” The study shows that when the complex basis is extended to prime fractions and their operator sequences, complex numbers are no longer isolated static coordinates but fractal topological operators with infinitely nested characteristics. This system re-establishes “generalized factorization” on the continuous complex plane, opening a new path for the control of complex dynamical systems, topological cryptography, and the geometrization of algebraic number theory.

Keywords: prime fractions; complex plane; Gaussian integers; generalized factorization; fractal topology

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I. Introduction: The Historical Compromise of Gaussian Integers

Since Gauss introduced the “Gaussian integers (a+bi,\ a,b\in\mathbb{Z}),” algebraic number theory has been able to continue the fundamental theorem of arithmetic on the complex plane. However, the price of this system is forced “discretization.” Because a,b are restricted to integers, the continuous complex plane is cut into isolated lattice points, losing the continuity most valuable to analysis and fractal geometry.

The reason traditional mathematicians did not dare to let a,b become fractions is that a continuous domain admits infinite subdivision, which would cause factorization to fail completely. However, under the FCFG system, by introducing “prime-fraction operators (1/p),” we have found a new path that preserves continuity while enabling topological deconstruction.

II. Prime-Fraction Complex Topology: From Static Points to Dynamic Operators

2.1 The Geometric Meaning of Prime-Fraction Bases

When the real part a and imaginary part b of a complex number z=a+bi are extended to prime fractions (e.g., a=1/p,\ b=1/q), the complex number z undergoes an essential geometric leap: it is no longer a flat coordinate point but a “rotation-scaling” topological operator. The real part 1/p determines the microscopic frequency band of the inward collapse of space, while the imaginary part 1/q determines the fractal folding density of the phase.

2.2 The Birth of the Fractal Complex Plane

The set of complex numbers generated by prime-fraction operators breaks the homogeneous texture of the traditional complex plane. The entire complex plane evolves into a topological manifold composed of infinitely nested miniature universes. Every prime-fraction coordinate carries within itself an infinitely magnifiable self-similar structure.

III. Reconstruction of Generalized Factorization on the Complex Plane

Traditional factorization fails on the continuous complex plane because a continuous domain admits infinite subdivision. However, after introducing prime-fraction operators, generalized factorization becomes valid on the complex plane: any complex composite complex-fractal manifold can be reverse-deconstructed into a specific set of “prime-fraction rotation-scaling operator sequences.”

A complex complex-topological structure can be uniquely represented as:

Z=\left(\frac{1}{p_1}+\frac{1}{q_1}i\right)\otimes

\left(\frac{1}{p_2}+\frac{1}{q_2}i\right)\otimes\cdots

This marks a formal upgrade of the object of study on the complex plane from “numerical computation” to “the generation and deconstruction of topological structures.”

IV. A Dimensional-Reduction Strike on Complex Dynamical Systems

In classical complex dynamical systems such as the Mandelbrot set, small changes in parameters often lead to chaos. After introducing prime-fraction complex topology, we can use “prime-fraction frequencies” to precisely lock onto and control fractal morphology.

For example, in the iteration equation z_{n+1}=z_n^2+c, when the constant c takes specific prime-fraction operators, the system will generate “native complex fractals” with strong regularity and self-similarity. This provides a foundational mathematical tool for signal modulation and system control in complex chaotic systems.

V. Topological Cryptography: A “Geometric Lock” in the Continuous Complex Plane

Traditional cryptography relies on discrete domains. Prime-fraction complex topology offers a completely different encryption possibility: encoding confidential data as prime-fraction recursive sequences of complex coordinates. Decryption requires not only arithmetic operations but also the complete topological deconstruction parameters. Because existing computational models are not good at reverse engineering fractal topology in continuous domains, this provides a high-dimensional security moat for geometric cryptography in the post-quantum era.

VI. Conclusion

The significance of prime fractions for complex numbers lies in removing the “integer shackles” imposed on complex numbers since Gauss. The complex plane is no longer merely the root space of algebraic equations but an infinitely nested geometric ocean driven by prime-fraction operators. The successful establishment of generalized factorization on the continuous complex plane not only broadens the research boundaries of number theory but also leaves an extremely broad space for exploration in future complex system analysis, continuous-domain cryptography, and generative geometry.