A Special Case of Dirichlet's Theorem on Primes in an Arithmetic Progression

Abstract

Dirichlet's theorem asserts that every arithmetic progression m, m + n, m + 2n, . . .. with m and n relatively prime, contains infinitely many primes. The simplest proofs are analytic, using properties of Dirichlet L-series [1], although Atle Selberg gave a complicated elementary proof in 1949 [5]. Certain individual cases, such as 3, 3 + 4, 3 + 8, ... and 5, 5 + 6, 5 + 12, ..., are easy to prove. Other special cases, notably 1, 1 + 4, 1 + 8, . . ., can be proved using simple properties of quadratic residues II]. In this note, we use elementary arguments to cover an infinite number of cases. While these have been given other elementary proofs (see, for instance, Dickson [2, vol. 1, p. 418] or Ribenboim [4, p. 268]), the proof presented here is the simplest and shortest that the author knows. In his recent paper in this MAGAZINE [3], Lionel Levine proved the following interesting theorem: