New Results for Covering Systems of Residue Sets

We announce some new results about systems of residue sets. A residue set R C Z is an arithmetic progression R = {a,a±n,a± 2n,...}. The positive integer n is referred to as the modulus of R. Following Znam [21] we denote this set by a(n). We need several number-theoretic functions. p(ra)-the least prime divisor of a natural number ra, P(m)-the greatest prime divisor of ra, A(m)-the greatest divisor of m which is a power of a single prime: A(ra) = max{d G Z: d\m, d = p s , p prime}, /(ra) = Ylj=i s j(Pj ~ 1) + 1> where ra has the prime factorization ra = Si Si g(m) = rij-iU + X J) ~ Ei=i x 3 ~ !> where Z^k=o Pj Pj 2^k=o Pj and m has the above prime factorization, 1, which partition Z. The multiplicity of a modulus n = rik is the number of sets in D with that modulus. The multiplicity of D is the maximum multiplicity of its moduli. THEOREM 1. The multiplicity of any modulus n = rik is at least (1) mi = min A (-, r). ni^n \{n,ni)J The multiplicity of D is at least (2) m 2-N + 1,