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February 19, 2026Open Access

A Fixed Modular Classification of Factor Residues for Semiprimes Modulo 90 (revised)

Key Points

The aim is to establish a fixed modular classification of factor residues for semiprimes modulo 90, improving factorization methods.
Developed a modular classification framework for semiprimes N = p·q modulo 90.
Utilized digital-root constraints (mod 9) and last-digit constraints (mod 10) to filter residue pairs.
Reduced combinatorial possibilities from 576 to approximately 12 unique unordered pairs per residue.
Created Master Inversion Tables for all 24 residues.
Achieved a reduction of approximately 96–98% in the search space for residue pairs.
Identified exactly 24 ordered residue pairs corresponding to true residues for semiprimes.
Presented numerical verification examples to validate the classification framework.

Abstract

Version 2 (expanded and revised) — This preprint presents a deterministic modular classification of factor residues for semiprimes N = p·q (with distinct odd primes p, q > 5) modulo 90. The residue r = N mod 90 determines exactly 24 ordered residue pairs (a, b) from the fixed set C = (ℤ/90ℤ)×, with exactly one pair (up to order) corresponding to the true residues of p and q. Using digital-root (mod 9) and last-digit constraints (mod 10), the framework reduces the combinatorial search space to approximately 12 unique unordered pairs per r, achieving a reduction of ~96–98 % from 576 possible ordered pairs. The structure is universal, size-independent, and closed under multiplication (closed-circle property). Complete Master Inversion Tables for all 24 residues are provided, along with numerical verification examples. This offers a pre-computational filter for targeted verification in restricted factorization or semiprime generation.

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