Disproof of a conjecture of Jacobsthal

Abstract:

For any integer $n\geq 1$, let $j(n)$ denote the Jacobsthal function, and $\omega (n)$ the number of distinct prime divisors of $n$. In 1962 Jacobsthal conjectured that for any integer $r\geq 1$, the maximal value of $j(n)$ when $n$ varies over ${\mathbb N}$ with $\omega (n)=r$ is attained when $n$ is the product of the first $r$ primes. We show that this is true for $r\leq 23$ and fails at $r=24$, thus disproving Jacobsthal’s conjecture.