Polynomial Residues and Quadratic Capture: A QCM View on the Erdős–Straus Equation

Description This work explores a structural connection between QCM (Quadratic Capturing Method) and the Erdős–Straus equation through explicit polynomial calculations and divisibility mechanisms. QCM is a quadratic modular method defined on integers of the formN = 2k + c (with c odd), where capture corresponds to the exact divisibility of a quadratic residue generated by a deterministic iteration. Experimental evidence shows strong structural properties, including divisor completeness, hierarchical capture ordering, and polynomial representations of the dynamics. Inspired by these mechanisms, we examine the Erdős–Straus equation from a polynomial perspective. By rewriting the equation as a symmetric cubic identity, a natural polynomial residue emerges whose exact divisibility characterizes integer solutions. Explicit calculations show that known solution families can be interpreted as minimal polynomial corridors, analogous to the quadratic corridors observed in QCM. The Chinese Remainder Theorem appears implicitly in both settings: compatibility across divisors is enforced by global structure rather than constructed locally. The aim of this work is not to provide a proof of the Erdős–Straus conjecture, but to propose a QCM-inspired structural framework linking quadratic congruence dynamics, polynomial residues, and Diophantine divisibility constraints. This presentation emphasizes explicit calculations, structural parallels, and limitations, offering a unified conceptual viewpoint for further exploration.