In mathematics, a covering system (also called a complete residue system) is a collection
of finitely many residue classes whose union contains every integer.
Examples and definitions
The notion of covering system was introduced by Paul Erdős in the early 1930s.
The following are examples of covering systems:
A covering system is called disjoint (or exact) if no two members overlap.
A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).
A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.
The first two examples are disjoint.
The third example is distinct.
A system (i.e., an unordered multi-set)
of finitely many residue classes is called an -cover if it covers every integer at least times, and an exact -cover if it covers each integer exactly times. It is known that for each there are exact -covers which cannot be written as a union of two covers. For example,
is an exact 2-cover which is not a union of two covers.
The Mirsky–Newman theorem, a special case of the Herzog–Schönheim conjecture, states that there is no disjoint distinct covering system. This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.
Covering systems can be used to find primefree sequences, sequences of integers satisfying the same recurrence relation as the Fibonacci numbers, such that consecutive numbers in the sequence are relatively prime but all numbers in the sequence are composite numbers. For instance, a sequence of this type found by Herbert Wilf has initial terms
In this sequence, the positions at which the numbers in the sequence are divisible by a prime p form an arithmetic progression; for instance, the even numbers in the sequence are the numbers ai where i is congruent to 1 mod 3. The progressions divisible by different primes form a covering system, showing that every number in the sequence is divisible by at least one prime.
Boundedness of the smallest modulus
Paul Erdős asked whether for any arbitrarily large N there exists an incongruent covering system the minimum of whose moduli is at least N. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120; a suitable cover is 0(3), 0(4), 0(5), 1(6), 1(8), 2(10), 11(12), 1(15), 14(20), 5(24), 8(30), 6(40), 58(60), 26(120) ) D. Swift gave an example where the minimum of the moduli is 4 (and the moduli are in the set of the divisors of 2880). S. L. G. Choi proved that it is possible to give an example for N = 20, and Pace P Nielsen demonstrates the existence of an example with N = 40, consisting of more than congruences.
Systems of odd moduli
There is a famous unsolved conjecture from Erdős and Selfridge: an incongruent covering system (with the minimum modulus greater than 1) whose moduli are odd, does not exist. It is known that if such a system exists with square-free moduli, the overall modulus must have at least 22 prime factors.
- Soifer, Alexander (2009). The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators. With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau. New York: Springer. pp. 1–9. ISBN 978-0-387-74640-1. MR 2458293. doi:10.1007/978-0-387-74642-5.
- Choi, S. L. G. (1971). "Covering the set of integers by congruence classes of distinct moduli". Math. Comp. 25 (116): 885–895. MR 0297692. doi:10.2307/2004353.
- Nielsen, Pace P. (2009). "A covering system whose smallest modulus is 40". Journal of Number Theory. 129 (3): 640–666. MR 2488595. doi:10.1016/j.jnt.2008.09.016.
- Hough, Bob (2015). "Solution of the minimum modulus problem for covering systems". Ann. of Math. 181 (1): 361–382. MR 3272928. doi:10.4007/annals.2015.181.1.6.
- Guo, Song; Sun, Zhi-Wei (2005). "On odd covering systems with distinct moduli". Adv. Appl. Math. 35 (2): 182–187. MR 2152886. arXiv: . doi:10.1016/j.aam.2005.01.004.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. Problem Books in Mathematics (3rd ed.). New York, NY: Springer-Verlag. pp. 383–385. ISBN 0-387-20860-7. Zbl 1058.11001.