On a Simple Algebraic Formula for the Distribution of Prime Numbers

Abstract:The distribution of prime numbers remains one of the most central and fascinating domains in analytic number theory. While classical approaches heavily rely on complex continuous analysis, this paper introduces a novel, structurally minimalist algebraic framework to characterize the macroscopic distribution of prime and composite numbers. By constructing a system of simultaneous equations governed by probability conservation and asymptotic density, we derive a quadratic characteristic equation that inherently establishes a lower definition boundary at x≧e⁴~ 54.598..., effectively filtering out localized stochastic perturbations (On_a_Simpl... pp. 5, 14). Furthermore, we incorporate the concept of Gini Impurity from information science to provide a profound statistical interpretation of the system's mixture chaos, demonstrating that its entropy is asymptotically governed by the natural logarithmic scale ln x (On_a_Simpl... pp. 8, 15). Through rigorous mathematical derivations spanning 15 pages, we demonstrate that the sequence of numerator coefficients for the higher-order asymptotic correction terms precisely corresponds to the classical Catalan numbers (On_a_Simpl... pp. 12, 14-15). Our results bridge analytic number theory with informational combinatorics, offering a fresh, self-consistent, and highly accessible algebraic perspective on the structural patterns of prime numbers (On_a_Simpl... pp. 14-15).