We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod $k$) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.

@misc{item61,
title = {Prime numbers in certain arithmetic progressions},
author = {Ram Murty and Nithum Thain},
url = {http://projecteuclid.org/download/pdf{\_}1/euclid.facm/1229442627},
urldate = {2016-05-07},
abstract = {We discuss to what extent Euclid's elementary proof of the infinitude of primes can be modified so as to show infinitude of primes in arithmetic progressions (Dirichlet's theorem). Murty had shown earlier that such proofs can exist if and only if the residue class (mod {\$}k{\$}) has order 1 or 2. After reviewing this work, we consider generalizations of this question to algebraic number fields.},
comment = {},
year = 2001,
collections = {Fun maths facts,Integerology}
}