Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2022) and another new proof

Abstract

{\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.