AI 中文总结
该研究提出素数发条这一离散动力系统，无需显式模除等运算即可生成素数并表示整数，其大循环对应中国剩余定理，还可用于计算多种数论函数。
AI 中文摘要
数的表示方式会直接影响哪些算术结构易于被观察到。素数发条（The Prime Clockwork）是一种递归增长的离散动力系统：由一系列自主的双指针时钟构成，这些时钟由同一个+1信号驱动，系统不提供任何素数或素性标签。系统从空状态启动，每当现有时钟响起时，就添加一个周期为n的时钟；素数会在系统增长过程中内部生成。对于每个已安装的素数p，其秒读数R_p会按0,…,p-1的顺序推进，每次回到0时，分钟读数M_p（记录已完成的p周期数）就会加1。这些指针仅使用增量、比较、重置和进位操作，无需显式的mod或div运算。在时间n处，满足n=pM_p(n)+R_p(n)。赋值读数V_p(n)=ν_p(n)由局部生成：当秒计数器非零时，赋值读数为0（静默状态）；当p时钟响起时，赋值读数为当前分钟读数所指向的更早赋值加1。赋值向量以唯一素因子分解形式给出整数，其坐标在乘法和除法下做加减运算，可唯一表示每个正有理数；整除性对应弱分量序，唯一素因子分解在该表示中是自然的。有限秒数组构成笛卡尔积状态空间，其共同轨道在重复前会遍历所有联合状态，这个大循环（grand cycle）是中国剩余定理的序敏感动力对应物。这些相同坐标还可用于计算最大公约数（gcd）、最小公倍数（lcm）、完全幂、贝祖等式（Bézout's identity）以及欧拉 totient 函数。有理赋值水平可达到某些正代数无理数，但通常无法达到代数数。
英文摘要
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 $V_p(n)=ν_p(n)$ 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.
CommentsThis manuscript develops a discrete dynamical-system representation of elementary number theory and uses it to give alternative proofs of classical results; it does not claim new theorems. We therefore selected History and Overview (math.HO) as the primary classification and Number Theory (math.NT) as a cross-list
