Disproof of a conjecture of Jacobsthal

Abstract

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