# Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 B.C.--2017) and another new proof

• Published in 2012
In this article, we provide a comprehensive historical survey of 183 different proofs of famous Euclid's theorem on the infinitude of prime numbers. 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 8 subsections of Section 2 in dependence of the methods that are used in them. {\bf Related new 14 proofs (2012-2017) are given in the last subsection of Section 2.} In the next section, we survey mainly 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.

### BibTeX entry

@article{Euclidstheoremontheinfinitudeofprimesahistoricalsurveyofitsproofs300BC2017andanothernewproof,
title = {Euclid's theorem on the infinitude of primes: a historical survey of its  proofs (300 B.C.--2017) and another new proof},
abstract = {In this article, we provide a comprehensive historical survey of 183
different proofs of famous Euclid's theorem on the infinitude of prime numbers.
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 8 subsections of Section 2 in dependence of the methods that are
used in them. {\{}\bf Related new 14 proofs (2012-2017) are given in the last
subsection of Section 2.{\}} In the next section, we survey mainly 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.},
url = {http://arxiv.org/abs/1202.3670v3 http://arxiv.org/pdf/1202.3670v3},
year = 2012,
author = {Romeo Me{\v{s}}trovi{\'{c}}},
comment = {},
urldate = {2018-08-21},
archivePrefix = {arXiv},
eprint = {1202.3670},
primaryClass = {math.HO},
collections = {Easily explained,History,Lists and catalogues,The act of doing maths,About proof,Integerology}
}