质数与二次剩余是随机的吗？
Are prime numbers and quadratic residues random?
AI总结：
本文提出一种基于单轨迹动力学熵的新分析方法，解决单个确定性序列的随机性/复杂性问题，证明二次剩余具高复杂性而质数复杂性意外低，为质数与二次剩余的随机性研究提供新视角。
摘要：
在现代科学出版物中，各种数论构造中对随机性的应用屡见不鲜，V.I. Arnold、M. Katz、Ya.G. Sinai和T. Tao等著名学者的研究就是例证。遗憾的是，所有这些方法都依赖于各种启发式方法，尽管这些方法通常非常非平凡且优雅。本文提出一种新的分析方法，用于解决单个确定性序列的随机性/复杂性问题。该方法证明了二次剩余具有预期的高复杂性，而质数的复杂性却意外地低。从技术上讲，我们的方法基于单个轨迹动力学熵的新构造，该熵用于度量轨迹的复杂性，这与度量整个动力系统复杂性的经典Kolmogorov-Sinai熵和拓扑熵形成对比。
英文摘要：
Appeals to randomness in various number-theoretic constructions appear regularly in modern scientific publications. Such famous names as V.I. Arnold, M. Katz, Ya.G. Sinai, and T. Tao are just a few examples. Unfortunately, all of these approaches rely on various, although often very non-trivial and elegant, heuristics. A new analytical approach is proposed to address the issue of randomness/complexity of an individual deterministic sequence. This approach demonstrates the expected high complexity of quadratic residues and the unexpectedly low complexity in the case of prime numbers. Technically, our approach is based on a new construction of the dynamical entropy of a single trajectory, which measures its complexity, in contrast to classical Kolmogorov-Sinai and topological entropies, which measure the complexity of the entire dynamical system.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
