On Dirichlet's Theorem and Infinite Primes

It is shown that Dirichlet's theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring Z of the nonstandard integers is the image of an infinite prime. The ring Z is the completion of Z relative to the "natural" topology on Z. 1. Notation. Throughout this note A shall denote the natural numbers, Z the rational integers, and P the positive primes. We shall follow the approach of Machover and Hirschfeld, [2], in our use of nonstandard analysis. Thus U is to be a universal set containing N and *U will be a comprehensive [6, p. 446] enlargement of U. The nonstandard natural numbers *A can be expressed as *A=AuAœ where Nw is the set of infinite natural numbers. Similarly, *P=P\JPœ, Px the set of infinite primes. 2. Lemma. Let a, b be coprime integers. A necessary and sufficient condition that the sequence \a+bn\ in e N)contains infinitely many primes is that \a+bn\ be an infinite prime for some nonstandard natural number n.