Capacity, Abundance, and the Impossibility of Small-Period Dual Coverings for Brier Numbers
Year: 2026
Type: preprint
Abstract: This work presents an advanced structural and computational study of Sierpinski-Riesel covering systems for Brier numbers, establishing a core identity linking covering capacity to the classical abundancy index. It develops a precise gcd-based criterion that eliminates the need for integer factorization, enabling computational sweeps up to periods of 10^9. Key results include a constructive classification of the smallest covering multipliers, first-principles derivations of the classical constants (78557, 271129, 509203), and a proof of the impossibility of shared-period dual coverings up to period 719 using Fourier and binary-tree methods. The paper concludes with a self-contained Python verification script reproducing every numerical claim, ensuring full transparency and reproducibility.
Author: النصراوي, ضياء
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7202247356