A Study on the Prime Factor Fractal Dimension and Its Convergence

This study introduces a novel mathematical concept: the Prime Factor Fractal Dimension (PFFD), denoted as Dₙ.Inspired by fractal geometry and information theory, Dₙ quantifies the complexity of an integer’s prime factor distribution by mapping its logarithmic factor sequence into a geometric space and applying box-counting dimension analysis.The paper defines Dₙ formally, explores its behavior across different classes of integers (primes, prime powers, square-free numbers, highly composite numbers), and proposes a central conjecture: \lim_{n \to \infty} D_n = Cwhere C is a transcendental constant potentially related to ln(2), γ (Euler–Mascheroni constant), or 1 - \frac{1}{\zeta(2)} .This framework offers a new lens for understanding number structure, with potential applications in cryptography, algorithmic complexity, and interactive game design. The research also explores connections to classical number-theoretic functions such as Euler’s totient function \varphi(n) , the Mertens function M(n) , and the Riemann zeta function \zeta(s) .Keywords: Prime Factorization, Fractal Dimension, Number Theory, Information Entropy, Logarithmic Mapping, Box-Counting, Kolmogorov Complexity---