On Odd Covering Systems with Distinct Moduli
نویسندگان
چکیده
A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}s=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}s=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.
منابع مشابه
On Odd Covering Systems with Distinct
Abstract. A famous unsolved conjecture of P. Erdős and J. L. Selfridge states that there does not exist a covering system {as(mod ns)}ks=1 with the moduli n1, . . . , nk odd, distinct and greater than one. In this paper we show that if such a covering system {as(mod ns)}ks=1 exists with n1, . . . , nk all square-free, then the least common multiple of n1, . . . , nk has at least 22 prime divisors.
متن کاملOn 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 Self...
متن کاملInfinite Covering Systems of Congruences Which Don’t Exist
We prove there is no infinite set of congruences with: every integer satisfying exactly one congruence, distinct moduli, the sum of the reciprocals of the moduli equal to 1, the lcm of the moduli divisible by only finitely many primes, and a prime greater than 3 dividing any of the moduli.
متن کاملOn the Reducibility of Exact Covering Systems
There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split of a coarser ECS. However, an ECS admiting a maximal modulus which is divisible by at most two distinct primes, primely splits a coarser ECS. As a consequence, if all moduli of an ECS A are divisible by at most two distinct primes, then A is natural. That is, A can be formed by iteratively splitting...
متن کاملComposite Covering Systems of Minimum Cardinality
We write S(m, a) for the congruence class {n ∈ Z : n ≡ a (mod m)}. A covering system of congruences is a collection {S(m1, a1), S(m2, a2), . . . , S(mn, an)} with the property that ∪i=1S(mi, ai) = Z. Such a system is composite and incongruent if the moduli {mi : i = 1, . . . , n} are composite and distinct. We describe the composite incongruent covering systems of minimum cardinality, thus answ...
متن کامل