New necessary and sufficient conditions on (ai, mi) in order that x ≡ ai (mod mi) be a covering system

A covering system is a set of congruences x ≡ ai (mod mi), i = 1, … k, such that every integer satisfies at least one of them. necessary and sufficient condition in order that a given set of congruences x ≡ ai (mod mi) be a covering system is established. w that (4) are such conditions. act covering systems they are reduced to (5). nnection of these conditions to known ones such as those [3] based on Bernoulli polynomials and those [8] based on cosets of Zm1 × Zm2 × … × Zmk are studied.