The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic
Abstract
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 Rp advances through 0,…,p−1, and each return to zero increments the minutes reading Mp, 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=pMp(n)+Rp(n). The valuation readout Vp(n)=νp(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.
Related
- Truncated pretentious distances of multiplicative arithmetic functions
- ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP
- Representation and Extending structure of a Multiplicative Lie algebra
- Arithmetic Siegel--Weil formula on the product of modular curves
- Prime-Power Multiplicative Modular Monoid Morphisms
BibTeX
@misc{emmerich2026prime,
title = {{The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic}},
author = {Michael T. M. Emmerich},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03896},
eprint = {2609.03896},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03896},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Prime Clockwork: A Dynamic Representation of Modular and Multiplicative Arithmetic
AU - Michael T. M. Emmerich
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03896
UR - https://arxiv.org/abs/2609.03896
AB - 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,…,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)=ν_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.
ER -