The Eight Factor Doors: From Digit Tails to a Reduced FactorScape
Year: 2026
Type: preprint
Abstract: Every prime greater than 5, and every composite number relatively prime to 30, belongs to one of eight residue families. These families are ordinarily encountered as the survivors of a familiar sieve: numbers not divisible by 2, 3, or 5. This paper asks what becomes visible when they are treated instead as eight structured doors through which factors may enter a product. Beginning with the decimal tails 1, 3, 7, and 9, the eight families are organized into a reduced arithmetic landscape. Their multiplication relations separate the landscape into two regions; their square residues distinguish two exceptional families; and their admissible factor pairings generate explicit bilinear sheets of composite numbers. The same organization then passes into difference-of-squares factorization, where it restricts the possible centers of a Fermat search to selected lanes modulo 15. These lanes can also be translated into an additive gnomonic construction and a first-match construction based on descending squares. The result is not proposed as a new high-performance factorization algorithm. Its purpose is structural: to show that several apparently different procedures are coordinate views of one reduced FactorScape. In the spirit of WidiVision, the paper studies not only what arithmetic operations produce, but how a changed field of vision can reveal relationships among them.
Author: Widi, Bill
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212359003
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