Odd covering systems with moduli square-free at the primes up to 19

Abstract

Balister, Bollobás, Morris, Sahasrabudhe and Tiba proved that no finite collection of arithmetic progressions with distinct odd square-free moduli covers the integers, and observed that their proof uses square-freeness only at the primes p ≤ 73, asking to what extent 73 can be reduced. We reduce it to 19: a finite collection of arithmetic progressions with distinct odd moduli, none of which is divisible by p² for any prime p ≤ 19, does not cover the integers. The proof applies the removal of codimension-one progressions, which they use at the square-free primes, also at primes occurring to higher powers, and replaces their four-prime configuration analysis by an exact analysis of the primes 3, 5, 7, 11, 13 together, organised so that only the configurations that need it reach the fifth prime.