New computational results on a conjecture of Jacobsthal

TL;DR

This paper extends computational verification of Jacobsthal's conjecture, finding new counterexamples up to the product of 43 primes, suggesting the conjecture only holds for small values of k.

Contribution

We implemented an extended Greedy Permutation Algorithm and computed the maximum Jacobsthal function for larger k, providing new counterexamples and insights into the conjecture's limitations.

Findings

Disproved Jacobsthal's conjecture for larger k values

Identified patterns indicating the conjecture's validity only for small k

Provided exhaustive data on sequences related to the conjecture

Abstract

Jacobsthal's conjecture has been disproved by counterexample a few years ago. We continue to verify this conjecture on a larger scale. For this purpose, we implemented an extension of the Greedy Permutation Algorithm and computed the maximum Jacobsthal function for the product of $k$ primes up to $k=43$. We have found various new counterexamples. Their pattern seems to imply that the conjecture of Jacobsthal only applies to several small $k$. Our results raise further questions for discussion. In addition to this paper, we provide exhaustive information about all covered sequences of the appropriate maximum lengths in ancillary files.