AI 中文总结
研究从正整数\(n\)出发，在无先验信息时构造多项式商环，通过模指数运算恢复\((n,2n]\)内素数，仅用前一个素数就能得下一个素数，还发展了环结构、扩展结果并给出实现。
AI 中文摘要
从正整数\(n\)出发，在没有关于其上素数的先验信息的情况下，我们构造了一个多项式商环，通过一次模指数运算就能精确恢复\((n,2n]\)内的素数。素数作为所得多项式余数的非零单项式次数同时出现，每个系数通过其加法阶独立认证其对应的素数。当\(n = p_k\)为素数时，最小非零次数为\(p_{k + 1}\)。由此仅从前一个素数就能恢复下一个素数，无需使用索引\(k\)、素数计数函数、素数表或任何素性测试。我们发展了底层环结构，给出了等价的零化子和商形式，将结果扩展到更短区间，并提供了SageMath实现。
英文摘要
Starting from a positive integer $n$ and no a priori information about the primes above it, we construct a polynomial quotient ring that recovers exactly the primes in $(n,2n]$ from a single modular exponentiation. The primes occur simultaneously as the nonzero monomial degrees of the resulting polynomial remainder, and each coefficient independently certifies its corresponding prime through its additive order. When $n=p_k$ is prime, the least nonzero degree is $p_{k+1}$. Thus the next prime is recovered from the preceding prime alone, without using the index $k$, the prime-counting function, a prime table, nor any primality tests. We develop the underlying ring structure, give equivalent annihilator and quotient formulations, extend the result to shorter intervals, and provide a SageMath implementation.
