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.

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

References in corpus (1)

• Dirichlet's theorem and Jacobsthal's function

Cited by in corpus (1)

• On differences between consecutive numbers coprime to primorials