Dirichlet's theorem and Jacobsthal's function

TL;DR

This paper explores the relationship between Dirichlet's theorem on primes in arithmetic progressions and bounds on the Jacobsthal function, proposing that certain bounds could lead to an elementary proof of Dirichlet's theorem.

Contribution

It demonstrates that for moduli up to 76, eligible arithmetic progressions contain primes, and suggests that bounds on the Jacobsthal function could imply an elementary proof of Dirichlet's theorem.

Findings

Eligible arithmetic progressions with d ≤ 76 contain primes.

Certain bounds on the Jacobsthal function imply primes in all eligible progressions.

Potential for elementary proof of Dirichlet's theorem based on Jacobsthal function bounds.

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.