What question did this study set out to answer?

The aim is to explore geometrical structures in numbers of the form 6n ± 1 to improve factor exclusion methods.

February 12, 2026Open Access

Modular Geometric Structure and Factor Exclusion Principles

Key Points

The aim is to explore geometrical structures in numbers of the form 6n ± 1 to improve factor exclusion methods.
Developed a modular geometric structure for integers of the form 6n ± 1.
Created two sieving systems: Affine Modular Sieve (AMS) and Modular Circle Sieve (MCS).
Formulated the Geometric Factor Bounding Theorem (GFBT) and Factor Exclusion Theorem (FET).
Presented a deterministic factorization procedure based on modular geometry.
Established upper bounds on factors of composite integers via GFBT.
Introduced exclusion frameworks for factors based on congruential relations.
Noted structural asymmetry between the classes 6n + 1 and 6n − 1.
Illustrated the completeness of exclusion information through a case study.

Abstract

We develop a modular geometric structure for integers of the form 6n ± 1, leadingto two exact sieving systems: the Affine Modular Sieve (AMS) and the ModularCircle Sieve (MCS). From the structure of the MCS, we formulate two exclusionmechanisms operating at different levels. The first, the Geometric Factor BoundingTheorem (GFBT), establishes a local, deterministic upper bound on the smaller factorof a composite integer, contingent upon its residue class. The second, the FactorExclusion Theorem (FET), introduces a framework for excluding factor candidatesthrough geometrically induced congruential relations.In addition, we introduce a deterministic factorization procedure derived frommodular geometry, intended as a structural tool rather than an optimization method.An analysis of the associated search domain reveals a structural asymmetry betweenthe classes 6n + 1 and 6n − 1, situating the work within a unified conceptual approachto the geometric exclusion of factors. Finally, we present a case study that illustratesthe local completeness of the exclusion information encoded in the geometric intervals

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