Multifractal structure in prime number distribution: analysis of autorecurrence mechanisms

This work presents a numerical analysis of the autorecurrent equation x = a + a/x and searches for its potential manifestations in the distribution of prime numbers. It has been demonstrated that the golden ratio φ = 1.618... represents a special case solution for unit a = 1, rather than a universal constant. The developed Genesis Model, utilizing transformations through φ and local perturbations, achieves a high degree of effectiveness in prime number prediction. An atomic structure has been identified in the interval [1, 76] with center at L(8) = 47 and Genesis point at (φⁿ × 76). The analysis revealed regular structure of regions and rescaling points every 10 levels of the Lucas sequence (φ¹⁰ ≈ 123). Pattern investigation indicates the potential role of Euler's number e as a universal divisor. The Ω method for numerical integration and applications in cryptographic system analysis have been proposed. The presented hypotheses are speculative in nature and require systematic theoretical and empirical verification on broader datasets. The results suggest the possibility of deterministic mechanisms organizing prime numbers, arising from the projection of a universal autorecurrent process onto discrete structures. KEYWORDS: autorecurrence, prime numbers, golden ratio, discretization, cryptography, RSA, data compression, riemann hypothesis