Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes
Abstract
Purple Mathematics is an expository, unifying framework that gives every positive integer a structured identity through one local operation, the *dig*: factor n² − 1 = (n − 1)(n + 1) and read its largest prime factor ρ(n), comparing it to a fixed wall W = 11. This sorts the integers into a Zero domain and an Infinity domain along a "bridge" between 0 and infinity, with the unit 1 as the fixed point of x ↦ 1/x. The Zero domain is finite (Størmer's theorem): exactly 49 integers, the largest being 19,601. The paper develops the resulting structure, the choice of the wall, a triangular wall whose heights are the prime gaps, an identity card separating identity fields from diagnostic measurements, two independent signed calibrations, a structural-stability interpretation layer, an exact on-line detection criterion, and one genuinely open analytic problem about an associated prime-driven walk. The framework is explicitly a lens over classical results (Størmer, Dickman, prime gaps, Pólya–Liouville oscillation, twin-smooth integers); no new theorem is claimed, and every component is mapped to its established counterpart.
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BibTeX
@misc{safar2026purple,
title = {{Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes}},
author = {Samir Hanna Safar},
howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
year = {2026},
month = may,
doi = {10.5281/zenodo.20753251},
url = {https://doi.org/10.5281/zenodo.20753251},
note = {CC BY},
}
RIS
TY - UNPB
TI - Purple Mathematics: A Visual Framework for Numerical Identity, Determination, and the On-Line Detection of Primes
AU - Samir Hanna Safar
PY - 2026
DA - 2026/05/18
PB - Zenodo (CERN European Organization for Nuclear Research)
DO - 10.5281/zenodo.20753251
UR - https://doi.org/10.5281/zenodo.20753251
AB - Purple Mathematics is an expository, unifying framework that gives every positive integer a structured identity through one local operation, the *dig*: factor n² − 1 = (n − 1)(n + 1) and read its largest prime factor ρ(n), comparing it to a fixed wall W = 11. This sorts the integers into a Zero domain and an Infinity domain along a "bridge" between 0 and infinity, with the unit 1 as the fixed point of x ↦ 1/x. The Zero domain is finite (Størmer's theorem): exactly 49 integers, the largest being 19,601. The paper develops the resulting structure, the choice of the wall, a triangular wall whose heights are the prime gaps, an identity card separating identity fields from diagnostic measurements, two independent signed calibrations, a structural-stability interpretation layer, an exact on-line detection criterion, and one genuinely open analytic problem about an associated prime-driven walk. The framework is explicitly a lens over classical results (Størmer, Dickman, prime gaps, Pólya–Liouville oscillation, twin-smooth integers); no new theorem is claimed, and every component is mapped to its established counterpart.
ER -