On odd covering systems with distinct moduli

A famous unsolved conjecture of P. Erdős and J.L. Selfridge states that there does not exist a covering system {as(modns)}s=1k with the moduli n1,…,nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(modns)}s=1k exists with n1,…,nk all square-free, then the least common multiple of n1,…,nk has at least 22 prime divisors.