AI 中文总结
研究斯科伦问题，引入线性递归序列“大”零概念，通过定义“好素数”集，在加强的克拉默猜想条件下得出大零不存在可致斯科伦问题可判定，还无条件证明大零稀疏并得到通用斯科伦集。
AI 中文摘要
斯科伦问题是确定给定整数线性递归序列（LRS）是否有零项，其可判定性数十年来一直未决，在计算机科学诸多领域出现。本文引入（非退化）线性递归序列“大”零的概念，即出现在大于定义给定LRS数据大小的双指数的索引处的零。建立两个主要结果：一是定义无限素数集“好素数”，密度为1，对于给定LRS的任何大零，其周围区间及该区间内好素数数量有上限，在加强的克拉默猜想条件下，大零不存在，这将导致斯科伦问题可判定；二是无条件证明大零非常稀疏，能作为某些LRS大零的正整数集密度为零，进而得到密度为1的通用斯科伦集，回答了文献中一个未决问题。
英文摘要
The Skolem Problem asks to determine whether a given integer linear recurrence sequence (LRS) has a zero term. This problem, whose decidability has been open for many decades, arises across a wide range of topics in computer science, including loop termination, formal languages, automata theory, and probabilistic model checking, amongst many others. In the present paper, we introduce a notion of "large" zeros of (non-degenerate) linear recurrence sequences, i.e., zeros occurring at an index larger than a double exponential of the magnitude of the data defining the given LRS. We establish two main results. First, we define an infinite set of prime numbers, termed "good", having density one amongst all prime numbers, with the following property: for any large zero of a given LRS, there is an interval around the large zero together with an upper bound on the number of good primes possibly present in that interval. The bound in question is much lower than one would expect if good primes were distributed similarly as ordinary prime numbers, as per the Cramér model in number theory. We therefore conclude, conditionally on a strengthening of the classical Cramér conjecture, that large zeros do not exist, which would entail decidability of the Skolem Problem. Second, we show unconditionally that large zeros are very sparse: the set of positive integers that can possibly arise as large zeros of some LRS has null density. This in turn immediately yields a Universal Skolem Set of density one, answering a question left open in the literature.
