On differences between consecutive numbers coprime to primorials

TL;DR

This paper investigates the differences between consecutive numbers coprime to primorials, relating these differences to the Jacobsthal function, and provides bounds and computational data up to the 44th prime.

Contribution

It establishes bounds for even differences between coprimes to primorials and offers extensive computational results, advancing understanding of their distribution.

Findings

All even numbers below a certain bound occur as differences.

The Jacobsthal function bounds the maximum difference.

Computational data up to the 44th prime supports the conjecture.

Abstract

We consider the ordered sequence of coprimes to a given primorial number and investigate differences between consecutive elements. The Jacobsthal function applied to the concerning primorial turns out to represent the greatest of these differences. We will explore the smallest even number which does not occur as such a difference. Little is known about even natural numbers below the respective Jacobsthal function which cannot be represented as a difference between consecutive numbers coprime to a primorial. Existence and frequency of these numbers have not yet been clarified. Using the relation between restricted coverings of sequences of consecutive integers and the occuring differences, we derive a bound below which all even natural numbers are differences between consecutive numbers coprime to a given primorial $p_k\#$. Furthermore, we provide exhaustive computational results on non-existent differences for primes $p_k$ up to $k=44$. The data suggest the assumption that all even natural numbers up to $h(k-1)$ occur as differences of coprimes to $p_k\#$ where $h(n)$ is the Jacobsthal function applied to $p_n\#$.