Predictive Geometric Analysis of Even Perfect Numbers

Using the 52 known even perfect numbers, this paper identifies underlying structural regularities by projecting discrete arithmetic patterns into a continuous geometric space via a characteristic quadratic mapping. We establish that the macroscopic spacing between consecutive perfect numbers is explicitly governed by this univariate quadratic framework, reflecting an intrinsic algebraic-geometric structure driven by non-linear expansion rather than isolated arithmetic anomalies. Building upon the baseline 12k+7, a predictive analytic formulation is developed to quantify the spatial growth and arithmetic gaps of admissible candidates. The resulting methodology provides a robust pathway for estimating the magnitudes of higher-order perfect numbers, transcending the computational limitations of discrete primality testing. MSC Classification: Primary 11A25; Secondary 11A41 , 11N37 , 11N05