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| In [[mathematics]], a '''covering system''' (also called a '''complete residue system''') is a collection
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| :<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>
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| of finitely many
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| residue classes <math> a_i(\mathrm{mod}\ {n_i}) = \{ a_i + n_ix:\ x \in \Z \} </math>
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| whose union ''covers'' all the integers.
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| {{unsolved|mathematics|
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| For any arbitrarily large natural number ''N'' does there exist an incongruent covering system the minimum of whose moduli is at least ''N''?}}
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| == Examples and definitions ==
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| The notion of covering system was introduced by [[Paul Erdős]] in the early 1930s.
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| The following are examples of covering systems:
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| :<math>\{0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {3}),\ 2(\mathrm{mod}\ {3})\},</math>
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| and
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| :<math>\{1(\mathrm{mod}\ {2}),\ 2(\mathrm{mod}\ {4}),\ 4(\mathrm{mod}\ {8}),\ 0(\mathrm{mod}\ {8})\},</math>
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| and
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| :<math>\{0(\mathrm{mod}\ {2}),\ 0(\mathrm{mod}\ {3}),\ 1(\mathrm{mod}\ {4}),
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| \ 5(\mathrm{mod}\ {6}),\ 7(\mathrm{mod}\ {12})
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| \}.</math>
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| A covering system is called disjoint (or exact) if no two members overlap.
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| A covering system is called distinct (or incongruent) if all the moduli are different (and bigger than 1).
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| A covering system is called irredundant (or minimal) if all the residue classes are required to cover the integers.
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| The first two examples are disjoint.
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| The third example is distinct.
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| A system (i.e., an unordered multi-set)
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| :<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>
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| of finitely many
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| residue classes is called an <math>m</math>-cover if it covers every integer at least
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| <math>m</math> times, and an exact <math>m</math>-cover if it covers each integer exactly <math>m</math> times. It is known that for each
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| <math>m=2,3,\ldots</math> there are exact <math>m</math>-covers which cannot be written as a union of two covers. For example,
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| :<math>\{1(\mathrm{mod}\ {2});\ 0(\mathrm{mod}\ {3});\ 2(\mathrm{mod}\ {6});\ 0,4,6,8(\mathrm{mod}\ {10});
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| </math>
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| :<math>1,2,4,7,10,13(\mathrm{mod}\ {15});\ 5,11,12,22,23,29(\mathrm{mod}\ {30})
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| \}</math>
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| is an exact 2-cover which is not a union of two covers. | |
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| == Mirsky–Newman theorem ==
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| 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]].<ref name="soifer">{{citation
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| | last = Soifer | first = Alexander | author-link = Alexander Soifer
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| | isbn = 978-0-387-74640-1
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| | location = New York
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| | publisher = Springer
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| | title = The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators
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| | year = 2008
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| | contribution = Chapter 1. A story of colored polygons and arithmetic progressions
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| | pages = 1–9}}.</ref> | |
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| ==Primefree sequences==
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| Covering systems can be used to find [[primefree sequence]]s, sequences of integers satisfying the same [[recurrence relation]] as the [[Fibonacci number]]s, such that consecutive numbers in the sequence are [[relatively prime]] but all numbers in the sequence are [[composite number]]s. For instance, a sequence of this type found by [[Herbert Wilf]] has initial terms
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| :''a''<sub>1</sub> = 20615674205555510, ''a''<sub>2</sub> = 3794765361567513 {{OEIS|id=A083216}}. | |
| 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 ''a<sub>i</sub>'' 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.
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| == Some unsolved problems ==
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| [[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<ref>{{cite journal | author=S. L. G. Choi | title=Covering the set of integers by congruence classes of distinct moduli | journal=[[Math. Comp.]] | volume=25 | year=1971 | pages=885–895 | doi=10.2307/2004353 | jstor=2004353 | issue=116 | publisher=Mathematics of Computation, Vol. 25, No. 116 }}</ref> that it is possible to give an example for ''N'' = 20, and Pace P Nielsen demonstrates<ref>{{cite journal | author=Pace P. Nielsen | title=A covering system whose smallest modulus is 40 | journal=[[Journal of Number Theory]] | volume=129 | year=2009 | pages=640–666 | url=http://www.wiskundemeisjes.nl/wp-content/uploads/2009/02/sdarticle.pdf | doi=10.1016/j.jnt.2008.09.016 | issue=3}}</ref> the existence of an example with ''N'' = 40, consisting of more than <math>10^{50}</math> congruences.
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| It appears that this problem was solved negatively by Bob Hough, who talked about his work at the Erdős centennial conference in Budapest on July 3 2013.<ref>{{cite arXiv |eprint= 1307.0874 |author1= Bob Hough |title= The least modulus of a covering system |class= math.NT |year= 2013 }}</ref> He used the [[Lovász local lemma]] to show that there is some maximum ''N'' which can be the minimum modulus on a covering system. The proof is in principle [[Effective results in number theory|effective]], though an explicit bound is not given.
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| In another problem we want that all of the moduli (of an incongruent covering system) be odd. There is a famous unsolved conjecture from Erdős and [[John Selfridge|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.<ref>{{cite journal | title=On odd covering systems with distinct moduli | author=Song Guo | coauthors=Zhi-Wei Sun | date=14 September 2005 | volume=35 | issue=182 | journal=Adv. Appl. Math. , --187 | arxiv=math/0412217 }}</ref>
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| ==See also==
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| * [[Chinese remainder theorem]]
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| * [[Covering set]]
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| * [[Residue number system]]
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| ==References==
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| {{reflist}}
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| * {{cite book|author=Richard K. Guy|authorlink=Richard K. Guy|title=[[Unsolved Problems in Number Theory]]|publisher=[[Springer-Verlag]]|year=2004|isbn=0-387-20860-7|pages=383–385| zbl=1058.11001 | series=Problem Books in Mathematics | location=New York, NY }}
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| ==External links==
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| * [[Zhi-Wei Sun]]: [http://math.nju.edu.cn/~zwsun/Cover.pdf Problems and Results on Covering Systems] (a survey) ([[PDF]])
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| * Zhi-Wei Sun: [http://math.nju.edu.cn/~zwsun/Cref.pdf Classified Publications on Covering Systems] (PDF)
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| [[Category:Number theory]]
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| [[Category:Unsolved problems in mathematics]]
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