On the Infinitude of Covering Systems with Least Modulus Equal to 2

A finite set of residue classes ai (mod ni) with 1 < n1 < n2 < · · · < ns is called a covering system of congruences if every integer satisfies at least one of the congruences x ≡ai (mod ni). An example is the set {0 (mod 2), 1 (mod 3), 3 (mod 4), 5 (mod 6), 9 (mod 12)}. A covering system all of whose moduli are odd called an odd covering system is a famous unsolved conjecture of Erdös and Selfridge. In this paper, we establish that there exist infinitely many even covering systems in which the least modulus is 2 and all other moduli are even. In each such even covering system, the number of the moduli and their prime factors are determined. Moreover, we construct a covering system with nine moduli, the smallest modulus is 2, and the lcm of the moduli is divisible by only the primes 2 and 5. With the smallest modulus 2, this is an attempt in the direction of constructing covering systems none of whose moduli is a product of the prime 3.