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The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic — arXiv:2609.03896

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.03896
License (as archived): CC0-1.0
Archived: 2026-10-17T02:49:01.393Z
SHA-256 of the archived text: 681214c27381cefc68ce76db7229e0bc4d903d0c450dcf8d0f7652cf96a738dc
Stable URL of this page: https://metraipolis.com/documento/dc_0fca5e516dec

How to cite

The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic — arXiv:2609.03896 — arXiv (en), https://arxiv.org/abs/2609.03896 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_0fca5e516dec (Metraipolis archive dc_0fca5e516dec, sha256 681214c27381cefc68ce76db7229e0bc4d903d0c450dcf8d0f7652cf96a738dc)

BibTeX

@misc{dc0fca5e516dec,
  title = {The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic — arXiv:2609.03896},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.03896},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_0fca5e516dec; sha256 681214c27381cefc68ce76db7229e0bc4d903d0c450dcf8d0f7652cf96a738dc},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.03896. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The way numbers are represented strongly influences which arithmetic structures are easy to see. The \emph{prime clockwork} is a recursively growing discrete dynamical system: a list of autonomous two-hand clocks driven by one common $+1$ signal. No primes or primality labels are supplied. Starting empty, the process appends a clock of period $n$ whenever none already present rings; the primes are generated internally as its growth times. For each installed prime $p$, the seconds reading $R_p$ advances through $0,\ldots,p-1$, and each return to zero increments the minutes reading $M_p$, which counts completed $p$-cycles. The hands use only increment, comparison, reset, and carry, without explicit \texttt{mod} or \texttt{div} operations. At time $n$, $n=pM_p(n)+R_p(n)$. The valuation readout $V_p(n)=\nu_p(n)$ is generated locally: it is zero when the seconds counter is non-zero (silent state) and otherwise (when the p-clock rings) one plus the earlier valuation addressed by the current minutes reading. The valuation vector gives the integer in unique prime-factorized form. Its coordinates add and subtract under multiplication and division, representing every positive rational uniquely; divisibility becomes weak componentwise order, and unique factorization is natural in this representation. Finite seconds arrays form Cartesian-product state spaces whose common orbit visits every joint state once before repeating; this \emph{grand cycle} is the order-sensitive dynamical counterpart of the Chinese remainder theorem. The same coordinates expose gcd, lcm, perfect powers, Bézout's identity, and Euler's totient. Rational valuation levels reach certain positive algebraic irrationalities, but not algebraic numbers in general.

Tema degli esploratori: storia e filosofia della scienza (retro-catalogo arXiv: annunci del 2026-09-04 in math.HO).
Categorie arXiv: History and Overview (math.HO) ; Discrete Mathematics (cs.DM); Dynamical Systems (math.DS); Number Theory (math.NT); Cellular Automata and Lattice Gases (nlin.CG).
Pagina del lavoro: https://arxiv.org/abs/2609.03896 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T02:26:44.964Z dalla pagina https://arxiv.org/catchup/math.HO/2026-09-04?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).