Enots Wolley 序列的满射性
Surjectivity of the Enots Wolley Sequence
AI总结：
本文证明 Enots Wolley 序列包含所有具有至少两个不同素因子的正整数，通过反证法结合连续段分析、素数交换构造与不相交覆盖论证，仅依赖素数定理和梅滕斯估计，确立了序列的满射性。
AI中文摘要：
我们证明了 Enots Wolley 序列包含所有具有至少两个不同素因子的正整数。假设某个符合条件的整数被遗漏，并设 T 为其有限素因子集合。存在一个有限截断值，使得任何从该截断之后开始、且各项能被 T 中至少一个素数整除的极大连续段，其首项必能被 T 中所有素数整除，且该连续段的长度至多为 2。若此类连续段出现无穷多次，则能被 T 中部分（而非全部）素数整除的项数，至多比能被 T 中全部素数整除的项数多出一个固定常数。一个素数交换构造在任意大的尺度上给出相反的不等式：在剔除一个可忽略的例外集后，加权双重计数表明前一类项数存在固定比例的超出。因此，此类连续段只能出现有限多次。素数递推进而迫使所有足够晚的项都能被 T 中某个素数整除。最后，一个不相交覆盖论证排除了任何有限的最终素数覆盖，从而证明了满射性。所涉及的解析数论输入仅为素数定理和关于倒数素数的梅滕斯估计。
英文摘要：
We prove that the Enots Wolley sequence contains every positive integer with at least two distinct prime divisors. Suppose, toward a contradiction, that some eligible integer is omitted, and consider its finite set of prime divisors. The local rules then severely restrict how terms involving these primes can occur: after a finite initial segment, terms divisible by some but not all of them can outnumber terms divisible by all of them by at most a fixed constant. A prime-exchange construction gives the opposite conclusion at large scales. From almost every term divisible by all of the chosen primes, it produces enough smaller earlier terms divisible by only some of them; a weighted double count makes this excess quantitative and yields a contradiction. It follows that any omission would force every sufficiently late term to have a prime divisor in one fixed finite set. Prime recurrence and a disjoint-cover argument rule out such a finite obstruction, proving surjectivity. The only analytic number-theoretic inputs are the prime number theorem and Mertens' estimate for reciprocal primes.
