New computational results on a conjecture of Jacobsthal

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.

arXiv comment14 pages, 1 figure, 2 tables; v2: paper unchanged, redundant column removed from ancillary file permutations_c.txt

References in corpus (2)

• Algorithmic concepts for the computation of Jacobsthal's function
• Dirichlet's theorem and Jacobsthal's function

Cited by in corpus (1)

• On differences between consecutive numbers coprime to primorials