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The 6k ±1 Composite–Displacement Sieve: Formal Analysis and Efficient Implementation | Synapse

February 9, 2026Open Access

The 6k ±1 Composite–Displacement Sieve: Formal Analysis and Efficient Implementation

Key Points

To rigorously analyze the Composite–Displacement sieve and its applications in prime factorization.
Developed an ordered wheel factorization restricted to the residue classes 6k±1.
Utilized Selberg's combinatorial sieve with unit weights.
Employed Buchstab's delay-differential framework to analyze the output.
Proved that the resulting bitmap is exactly the Selberg remainder.
Demonstrated that the sieve inherits Buchstab densities.

Abstract

We give a rigorous treatment of the Composite–Displacement (“Greedy–Ast”) sieve, an or- dered wheel factorisation restricted to the two residue classes 6k±1. Building on Selberg’s combinatorial sieve with unit weights and Buchstab’s delay–differential framework, we prove that the resulting bitmap is exactly the Selberg remainder and inherits Buchstab densities.

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Cite This Study

Stephen Steiner (Tue,) studied this question.

synapsesocial.com/papers/69897a25f0ec2af6756e8721 https://doi.org/https://doi.org/10.5281/zenodo.18518119