分圆素数提取器
Cyclotomic Prime Extractors
AI总结：
本文提出从分圆多项式值恢复素数的显式公式，利用二进制整除模式和小修正识别素因子，并定义常数 $\Omega$ 以递归恢复所有奇素数。
AI中文摘要：
我们开发了从分圆多项式值 $\Phi_n(2)$ 中恢复素数的显式公式。二进制整除模式检测重复的素因子，并识别无平方因子指标的最小素因子，而对 $\log_2\Phi_n(2)$ 的小修正允许连续恢复不同的素因子。将指标特殊化，可得到素数乘积和大于给定整数的最小素数的恒等式。同一机制从有限分解扩展到无限素数序列：沿奇数原初数的分圆值的归一化极限定义了一个实常数 $\Omega=0.25061403238015047218\ldots$，通过递归舍入规则可从该常数恢复每个奇素数。
英文摘要：
We develop explicit prime-recovery formulas from the values $Φ_n(2)$ of cyclotomic polynomials. Binary divisibility patterns detect repeated prime factors and identify the least prime divisor of a squarefree index, while small corrections to $\log_2Φ_n(2)$ allow successive recovery of the distinct prime factors. Specializing the index gives identities for prime products and the least prime above a given integer. The same mechanism extends from finite factorizations to an infinite prime sequence: a normalized limit of cyclotomic values along the odd primorials defines a real constant $Ω=0.25061403238015047218\ldots$, from which every odd prime can be recovered by a recursive rounding rule.
