Covering Systems

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E0002/: Asks whether finite distinct covering systems can have arbitrarily large minimum modulus; Hough proved an absolute bound.

E0007/: Asks whether there is a covering system of congruences whose moduli are distinct and all odd.

E0008/: Asks whether every finite coloring of the integers admits a covering system whose moduli all receive the same color.

E0027/: Asks whether one constant C lets every tolerance and every N admit an almost covering system with distinct moduli all between N and C times N.

E0202/: The largest number of congruence classes with distinct moduli at most N that can be chosen so that no integer lies in two of them.

E0203/: Asks whether some integer coprime to 6 makes every number of the form two to the k times three to the l times it, plus one, composite.

E0204/: Asks whether some integer has a covering system using its divisors above one whose classes overlap only for coprime pairs of moduli.

E0273/: Asks whether there is a covering system of congruences whose moduli are all of the form p minus one for primes p at least 5.

E0274/: Asks whether a group can be exactly covered by more than one coset when the cosets have different sizes, each element lying in exactly one.

E0275/: Asks whether a system of r congruences that covers two to the r consecutive integers must cover every integer.

E0276/: Asks whether some Fibonacci-type sequence has all terms composite while no integer shares a factor with every term.

E0277/: Asks whether, for every c, some n has sum of divisors above c times n yet admits no covering system whose moduli are distinct divisors of n above one.

E0278/: The maximum density of integers covered by choosing one congruence class for each modulus in a finite set, and whether equal classes minimize the density.

E0279/: Asks whether, for every k at least 3, congruence classes can be chosen modulo each prime so that all large integers lie in one of them with quotient at least k.

E0280/: Asks whether a sequence of moduli growing faster than k log k always leaves a number of uncovered integers below each modulus that is not small compared with k.

E0281/: Asks whether moduli whose congruences always leave a density-zero set must have a finite initial segment leaving density below any given epsilon.

E0586/: Asks whether there is a system of congruences covering all integers in which no modulus divides another.

E0947/: Asks whether an exact covering system exists, finitely many congruence classes with distinct moduli such that every integer satisfies exactly one of them.

E1113/: Concerns Sierpinski numbers, odd m for which two to the k times m plus one is never prime, and the sets of primes that divide all those values.

E1188/: Concerns covering systems with distinct moduli that are minimal, in that no proper subsystem still covers every integer.

E1189/: Concerns sets of distinct moduli that can cover the integers by some choice of residues but have no proper subset that can.

E1190/: Asks for the size of the supremum of reciprocal sums over finite disjoint congruence families with distinct moduli greater than m; the supremum is known to logarithmic scale.

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Systems of congruences covering the integers, exact and disjoint coverings, the minimum modulus problem, and the analogous exact coverings of a group by cosets.

Site tags routed here: covering systems, divisors, group theory, number theory, primes.