A short note on Jacobsthal's function

TL;DR

This paper presents a new explicit upper bound for Jacobsthal's function, which measures the minimal length of integer sequences needed to guarantee a coprime integer to n.

Contribution

It provides a novel explicit upper bound on Jacobsthal's function, improving understanding of its growth and behavior.

Findings

Derived a new explicit upper bound for g(n)

Enhanced theoretical understanding of Jacobsthal's function

Potential applications in number theory and related fields

Abstract

The function g(n) represents the smallest number Q such that every sequence of Q consecutive integers contains an integer coprime to n. We give a new and explicit upper bound on this function.