Prime numbers in certain arithmetic progressions

Abstract

We discuss to what extent Euclid's elementary proof of the inflnitude of primes can be modifled so as to show inflnitude 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 flelds.