Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2012) and another new proof

摘要

In this article, we provide a comprehensive historical survey of differentproofs of famous Euclid's theorem on the infinitude of prime numbers. TheBibliography of this article contains 99 references consisting of 24 textbooksand monographs, 73 articles (including 20 {\it Notes} published in {\it Amer.Math. Monthly} and a few unpublished works that are found on Internet Websites,especially on {\tt http:arxiv.org/}), one {\it Ph.D. thesis} and {\it Sloane'sOn-Line Encyclopedia of Integer Sequences}. The all references concerning toproofs of Euclid's theorem that use similar methods and ideas are exposedsubsequently. Moreover, in Appendix we present a list of all 70 differentproofs of Euclid's theorem presented here together with the correspondingreference(s), the name(s) of his (their) author(s) and the main method(s)and/or idea(s) used in it (them). This list is arranged by year of publication. In Section 2, we give a new simple proof of the {\it infinitude of primes}.The first step of our proof is based on Euclid's idea. The remaining of theproof is quite simple and elementary and it does not use the notion ofdivisibility.