Mohammed Khalil Prime Sieve: The R' Series Algorithm

This paper introduces a novel, deterministic sieve methodology for prime number identification, founded on a 6x 1 matrix structure that effectively confines all prime candidates to two columns (C1 and C5). Within this streamlined structure, the study identifies distinct, stable periodic patterns governing the row and column coordinates of composite numbers. This insight allowed for the initial derivation of four fundamental positional formulas (R Series). Subsequently, these formulas were algebraically optimized into the final R' Series (R'1, R'2, R'3, R'4) to feature a unified structure that incorporates the x2 and xy terms. This unparalleled level of positional precision fundamentally transforms prime identification from an iterative elimination process into a direct, deterministic positional mapping. The primary quantitative result is that the algebraic optimization substantially reduces the overall computational redundancy and overhead compared to the foundational equations, laying the groundwork for a highly efficient sieving algorithm. By directly locating every composite position, the remaining un-generated cells definitively correspond to prime numbers, offering fresh insights and significant potential computational advantages over traditional factorization-based sieves.