report · Zenodo (CERN European Organization for Nuclear Research)
This work compiles ten research notes alongside experimental measurements in computational number theory. It combines reviews of classical mathematical tools, including Brun and Selberg sieves and the Hardy-Littlewood singular series, with an original experimental measurement notebook. The investigation fills four empty cells in an arithmetic table, extends the Sato-Tate spectrometer to complex multiplication by Z[ω], and analyses the historical ROCA vulnerability period spanning 2016 to 2017. All reported numerical values are provided alongside reproducible, dependency-free Python source code. The measurements evaluate prime-pair gaps up to 246 across 123 families based on the Zhang-Maynard-Tao-Polymath framework, with data extended up to 10 to the power of 8. In addition, an analysis of 2,608 keys across three historical epochs for the ROCA vulnerability yielded zero flags, providing empirical verification without claiming formal mathematical proofs.
Empirical testing in number theory helps bridge the gap between abstract mathematical concepts and computational practice. By providing fully reproducible code and precise numerical measurements, this work offers verifiable data on prime distributions and cryptographic key characteristics. Such empirical baselines assist researchers in validating theoretical models and checking historical cryptographic implementations for known structural vulnerabilities without relying on unsupported theoretical leaps.
The primary real-world application connects to cryptographic auditing and security verification, particularly concerning the detection of historical ROCA vulnerabilities in cryptographic keys. Potential users include security auditors, cryptographic software developers, and academic researchers examining prime generation. Because the work consists of an exploratory experimental laboratory notebook without formal proofs, it represents early-stage research that requires further development before integration into commercial cybersecurity verification tools.
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This volume gathers ten notes written between 16 and 31 August 2026, extended by experimental measurements performed on 4–5 September 2026. It is organised in two parts: classical background (Brun's and Selberg's sieves, the Hardy–Littlewood singular series, Brahmagupta's identity, the LTE lemma), cited to their authors; and an original measurement notebook filling four empty cells of the arithmetic Mendeleev table, extending the Sato–Tate spectrometer to complex multiplication by Z[ω], and traversing the historical ROCA window (2016–2017). Every measurement is reproducible: dependency-free pure-Python source code is published alongside, and numerical constants are reported to the digit. We claim no proof of any open conjecture; we publish an honest laboratory notebook, in which the map predicts and the console confirms. v2 (7 September 2026): Addendum V — the complete family of 123 prime-pair gaps up to 246 (Zhang–Maynard–Tao–Polymath theorem; the gap-246 row itself measured at ratio 0.9999); measurements extended to 10^8; ROCA verdict extended to 2,608 keys across three epochs, 0 flags. Source code (reproducible): https://github.com/fouad-bensmail/congruential-audit Francais Ce recueil rassemble dix notes rédigées entre le 16 et le 31 août 2026, augmentées de mesuresexpérimentales réalisées entre le 4 et le 5 septembre 2026. Il est organisé en deux parties : desrappels classiques (crible de Brun et Selberg, série singulière de Hardy–Littlewood, identité deBrahmagupta, lemme LTE) cités à leurs auteurs, et un cahier de mesures originales portantsur quatre cases vides de la table de Mendeleïev arithmétique, sur l’extension du spectromètre deSato–Tate à la multiplication complexe par Z[ω], et sur la traversée de la fenêtre historique ROCA(2016–2017). Chaque mesure est reproductible : le code source Python pur (sans dépendance) estpublié en regard, et les constantes numériques sont rapportées au chiffre près. Nous ne prétendonsdémontrer aucune conjecture ouverte; nous publions un cahier de laboratoire honnête, où la carteprédit et la console confirme. v2 (7 September 2026): Addendum V — les123 familles de paires (Zhang–Maynard–Tao–Polymath, gap-246 mesuré à 0.9999); mesures étendues à 10^8; verdict ROCA élargi à 2,608 clées.
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DOI: 10.5281/zenodo.22314057
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