preprint · Zenodo (CERN European Organization for Nuclear Research)
This research presents three equivalent ways to understand Fermat's factorisation of a semiprime number N=pq. These include a trigonometric perspective involving a hidden angle, a Pythagorean approach using a second integer triangle, and a dyadic interpretation that links Fermat's method to the 2-adic grammar of Collatz sequences through a dyadic cascade. The study also defines the FIPS 186-4 compliance window in terms of an angle. Computational validation using a companion script shows a congruential potential holds for Sophie Germain primes, with high accuracy in measurement-to-prediction ratios. Furthermore, the work highlights a dyadic signature in the slopes of Cunningham chains. The authors clarify that this represents a change of coordinates rather than a practical attack, as sweeping the relevant angle remains computationally intensive for compliant cryptographic keys.
This work offers novel mathematical perspectives on number factorisation, a core concept in cryptography and number theory. By connecting different mathematical frameworks, it deepens our theoretical understanding of prime numbers and their properties, which could inform future cryptographic research and design.
The abstract indicates this research is a theoretical exploration of number factorisation methods, explicitly stating it is not an attack and that the process remains exponential for compliant cryptographic keys. Therefore, the abstract does not indicate a direct application pathway or immediate commercialisation potential.
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Addendum v4 à la Note Universelle VII. Trois lectures équivalentes de la factorisation de Fermat d'un semi-premier N = pq : trigonométrique (angle caché, sin θ = b/a), pythagoricienne (second triangle entier de cathète N, ψ = 2 arctan(b/a)), dyadique (la cascade N^{2^k} double l'angle et relie Fermat à la grammaire 2-adique de Collatz). La fenêtre de conformité FIPS 186-4 s'écrit φ > 2^{-100} rad. Numériquement, le script compagnon sg_crible.py valide le potentiel congruentiel e^{-S} = 2C2 sur les premiers de Sophie Germain (rapports mesure/prédiction 0.9917, 0.9981, 0.9993 à 10^6, 10^7, 10^8) et met en évidence la signature dyadique ln 2 de la carte n → 2n+1 dans les pentes des chaînes de Cunningham. Changement de coordonnées, pas attaque : balayer φ reste exponentiel sur clés conformes
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DOI: 10.5281/zenodo.22179572
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