Modular Factorization Pattern M.F.P and ECM

Abstract This article introduces five variants of the Pattern-Based Factorization Method (MFP), which combine structural formulations and heuristic techniques based on decimal digit distributions, modular arithmetic, and Pollard-Rho adaptations. Each variant is evaluated according to its efficiency, ability to detect true divisors, and applicability to large composite numbers—including RSA integers. The innovations include the use of decimal redistribution, structural GCD functions, adaptive Pollard increments, and an extended ECM method capable of extracting real divisors.