欧几里得素数无穷多定理：其证明的历史综述（公元前300年—2022年）以及另一个新证明
Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2022) and another new proof
AI总结：
本文是欧几里得素数无穷多定理证明的历史综述，梳理了公元前300年至2022年的200种证明，分类呈现不同时期、不同方法的证明，还给出了一个新的简洁欧几里得式证明。
摘要：
在本文的第四扩展版中，我们对著名的欧几里得素数无穷多定理的200种不同证明（公元前300年—2022年）进行了全面的历史综述。作者试图收集几乎所有已知的素数无穷多证明，包括一些可由已知问题或整除性质直接推导得出的证明。此外，本文还列出了不同等差数列中素数无穷多的众多初等证明。所有使用相似方法与思路的欧几里得定理证明的相关参考文献随后依次呈现。具体而言，根据证明所使用的方法，已呈现的证明被划分到第2节的前五个小节中。2012年至2017年期间被提出的14个证明在2.9小节中给出，2018年至2022年的18个最新证明在2.10小节中呈现。在第3节中，我们主要综述了不同等差数列中素数无穷多的初等证明，这些证明都是Dirichlet定理的特例。在第4节中，我们给出了一个新的简洁的素数无穷多的“欧几里得式证明”。
英文摘要：
{\bf In the fourth extended version of this article, we provide a comprehensive historical survey of 200 different proofs of famous Euclid's theorem on the infinitude of prime numbers (300 {\small B.C.}--2022)}. The author is trying to collect almost all the known proofs on infinitude of primes, including some proofs that can be easily obtained as consequences of some known problems or divisibility properties. Furthermore, here are listed numerous elementary proofs of the infinitude of primes in different arithmetic progressions. All the references concerning the proofs of Euclid's theorem that use similar methods and ideas are exposed subsequently. Namely, presented proofs are divided into the first five subsections of Section 2 in dependence of the methods that are used in them. 14 proofs which are proved from 2012 to 2017 are given in Subsection 2.9, and 18 recent proofs from 2018 to 2022 are presented in Subsection 2.10. In Section 3, we mainly survey elementary proofs of the infinitude of primes in different arithmetic progressions. Presented proofs are special cases of Dirichlet's theorem. In Section 4, we give a new simple ''Euclidean's proof'' of the infinitude of primes.
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