Dirichlet's theorem and Jacobsthal's function

Abstract

If $a$ and $d$ are relatively prime, we refer to the set of integers congruent to $a$ mod $d$ as an `eligible' arithmetic progression. A theorem of Dirichlet says that every eligible arithmetic progression contains infinitely many primes; the theorem follows from the assertion that every eligible arithmetic progression contains at least one prime. The Jacobsthal function $g(n)$ is defined as the smallest positive integer such that every sequence of $g(n)$ consecutive integers contains an integer relatively prime to $n$. In this paper, we show by a combinatorial argument that every eligible arithmetic progression with $d\le76$ contains at least one prime, and we show that certain plausible bounds on the Jacobsthal function of primorials would imply that every eligible arithmetic progression contains at least one prime. That is, certain plausible bounds on the Jacobsthal function would lead to an elementary proof of Dirichlet's theorem.