1576:. 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
1654:: 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.
637:
1095:
491:
1445:
1220:
254:
813:
140:
370:
1307:
1557:
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.
937:
1609:
686:
497:
1330:
957:
899:
879:
856:
836:
2012:
965:
688:
are different (and bigger than 1). Hough and
Nielsen (2019) proved that any distinct covering system has a modulus that is divisible by either 2 or 3.
1535:
1644:
376:
1338:
1101:
154:
714:
41:
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17:
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1993:
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1739:. With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau. New York: Springer. pp. 1–9.
1663:
1824:
1471:. However, Mirsky and Newman never published their proof. The same proof was also found independently by
1673:
1619:
1459:, states that there is no disjoint distinct covering system. This result was conjectured in 1950 by
1492:
904:
1935:
1890:
1702:
1488:
1546:
form an arithmetic progression; for instance, the even numbers in the sequence are the numbers
1748:
1542:
In this sequence, the positions at which the numbers in the sequence are divisible by a prime
632:{\displaystyle \{0{\pmod {2}},\ 0{\pmod {3}},\ 1{\pmod {4}},\ 5{\pmod {6}},\ 7{\pmod {12}}\}.}
1736:
The
Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of its Creators
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1945:
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1997:
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821:
2006:
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1460:
268:
146:
1693:
R. D. Hough, P. P. Nielsen (2019). "Covering systems with restricted divisibility".
1572:
there exists an incongruent covering system the minimum of whose moduli is at least
1880:
Hough, Bob (2015). "Solution of the minimum modulus problem for covering systems".
1668:
1508:
1476:
1090:{\displaystyle \{1{\pmod {2}};\ 0{\pmod {3}};\ 2{\pmod {6}};\ 0,4,6,8{\pmod {10}};}
1312:
This is just one case of the following fact: For every positive integer modulus
1974:
1926:
Guo, Song; Sun, Zhi-Wei (2005). "On odd covering systems with distinct moduli".
1904:
1782:
1464:
1233:
24:
1716:
1949:
1838:
1819:
1744:
486:{\displaystyle \{1{\pmod {2}},\ 2{\pmod {4}},\ 4{\pmod {8}},\ 0{\pmod {8}}\},}
1440:{\displaystyle \{0{\pmod {m}},\ 1{\pmod {m}},\ \ldots ,\ {m-1}{\pmod {m}}\}.}
1618:
Erdős's question was resolved in the negative by Bob Hough. Hough used the
1978:
1215:{\displaystyle 1,2,4,7,10,13{\pmod {15}};\ 5,11,12,22,23,29{\pmod {30}}\}}
249:{\displaystyle a_{i}{\pmod {n_{i}}}=\{a_{i}+n_{i}x:\ x\in \mathbb {Z} \},}
1989:
808:{\displaystyle \{a_{1}{\pmod {n_{1}}},\ \ldots ,\ a_{k}{\pmod {n_{k}}}\}}
135:{\displaystyle \{a_{1}{\pmod {n_{1}}},\ \ldots ,\ a_{k}{\pmod {n_{k}}}\}}
1778:"Covering the set of integers by congruence classes of distinct moduli"
1633:
1611:
congruences. Tyler Owens demonstrates the existence of an example with
1580:= 20, and Pace P Nielsen demonstrates the existence of an example with
959:-covers which cannot be written as a union of two covers. For example,
1940:
1795:
1777:
1707:
1895:
699:) if all the residue classes are required to cover the integers.
365:{\displaystyle \{0{\pmod {3}},\ 1{\pmod {3}},\ 2{\pmod {3}}\},}
1982:
1626:<10 which can be the minimum modulus on a covering system.
1641:
Does there exist a covering system with odd distinct moduli?
1228:
The first example above is an exact 1-cover (also called an
1530:
1232:). Another exact cover in common use is that of odd and
1225:
is an exact 2-cover which is not a union of two covers.
1650:
There is a famous unsolved conjecture from Erdős and
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157:
44:
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134:
1507:. For instance, a sequence of this type found by
1455:The Mirsky–Newman theorem, a special case of the
1820:"A covering system whose smallest modulus is 40"
1302:{\displaystyle \{0{\pmod {2}},\ 1{\pmod {2}}\}.}
274:The following are examples of covering systems:
267:The notion of covering system was introduced by
818:of finitely many residue classes is called an
8:
1491:, sequences of integers satisfying the same
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969:
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356:
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129:
45:
1990:Classified Publications on Covering Systems
1863:"A Covering System with Minimum Modulus 42"
838:-cover if it covers every integer at least
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881:-cover if it covers each integer exactly
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1979:Problems and Results on Covering Systems
1568:asked whether for any arbitrarily large
708:A system (i.e., an unordered multi-set)
1685:
1645:(more unsolved problems in mathematics)
1487:Covering systems can be used to find
702:The first two examples are disjoint.
7:
1503:but all numbers in the sequence are
259:whose union contains every integer.
1622:to show that there is some maximum
1561:Boundedness of the smallest modulus
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2013:Unsolved problems in number theory
14:
901:times. It is known that for each
18:Complete residue system modulo m
1636:Unsolved problem in mathematics
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1584:= 40, consisting of more than
1463:and proved soon thereafter by
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1528:= 3794765361567513 (sequence
932:{\displaystyle m=2,3,\ldots }
650:) if no two members overlap.
691:A covering system is called
653:A covering system is called
642:A covering system is called
1905:10.4007/annals.2015.181.1.6
1861:Owens, Tyler (2014-12-01).
1457:Herzog–Schönheim conjecture
1332:, there is an exact cover:
2029:
1717:10.1215/00127094-2019-0058
15:
1950:10.1016/j.aam.2005.01.004
1839:10.1016/j.jnt.2008.09.016
1818:Nielsen, Pace P. (2009).
1745:10.1007/978-0-387-74642-5
1664:Chinese remainder theorem
1825:Journal of Number Theory
263:Examples and definitions
16:Not to be confused with
1776:Choi, S. L. G. (1971).
1604:{\displaystyle 10^{50}}
33:complete residue system
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1674:Residue number system
1630:Systems of odd moduli
1606:
1521:= 20615674205555510,
1451:Mirsky–Newman theorem
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681:{\displaystyle n_{i}}
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665:
661:) if all the moduli
498:
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281:
271:in the early 1930s.
155:
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1867:BYU ScholarsArchive
1511:has initial terms
1493:recurrence relation
1489:primefree sequences
1483:Primefree sequences
1996:2007-09-29 at the
1620:Lovász local lemma
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35:) is a collection
1754:978-0-387-74640-1
1731:Soifer, Alexander
1701:(17): 3261–3295.
1505:composite numbers
1497:Fibonacci numbers
1403:
1394:
1369:
1325:{\displaystyle m}
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952:{\displaystyle m}
894:{\displaystyle m}
874:{\displaystyle m}
851:{\displaystyle m}
831:{\displaystyle m}
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1501:relatively prime
1473:Harold Davenport
1469:Donald J. Newman
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31:(also called a
29:covering system
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1969:External links
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711:
710:
709:
706:
703:
700:
698:
694:
689:
673:
669:
660:
656:
651:
649:
645:
626:
616:
612:
604:
598:
591:
587:
579:
573:
566:
562:
554:
548:
541:
537:
529:
523:
516:
512:
504:
494:
480:
470:
466:
458:
452:
445:
441:
433:
427:
420:
416:
408:
402:
395:
391:
383:
373:
359:
349:
345:
337:
331:
324:
320:
312:
306:
299:
295:
287:
277:
276:
275:
272:
270:
262:
260:
243:
232:
229:
223:
220:
215:
211:
207:
202:
198:
191:
182:
178:
173:
163:
159:
151:
150:
149:
148:
120:
116:
111:
101:
97:
90:
87:
81:
72:
68:
63:
53:
49:
38:
37:
36:
34:
30:
26:
19:
1981:(a survey) (
1941:math/0412217
1931:
1927:
1921:
1886:
1881:
1875:
1866:
1856:
1829:
1823:
1813:
1786:
1781:
1771:
1735:
1725:
1698:
1695:Duke Math. J
1694:
1688:
1669:Covering set
1649:
1623:
1617:
1612:
1581:
1577:
1573:
1569:
1564:
1554:
1547:
1543:
1541:
1522:
1515:
1509:Herbert Wilf
1486:
1477:Richard Rado
1454:
1311:
1234:even numbers
1229:
1227:
1224:
859:
817:
707:
704:
701:
696:
692:
690:
658:
654:
652:
647:
643:
641:
273:
266:
258:
144:
32:
28:
22:
1975:Zhi-Wei Sun
1783:Math. Comp.
1465:Leon Mirsky
1230:exact cover
693:irredundant
659:incongruent
25:mathematics
1708:1703.02133
1680:References
1566:Paul Erdős
1461:Paul Erdős
269:Paul Erdős
1896:1307.0874
1652:Selfridge
1409:−
1396:…
927:…
761:…
233:∈
88:…
2007:Category
1994:Archived
1733:(2009).
1658:See also
655:distinct
644:disjoint
1958:2152886
1913:3272928
1848:2488595
1805:0297692
1763:2458293
1534:in the
1531:A083216
1495:as the
697:minimal
1992:(PDF)
1956:
1911:
1846:
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1751:
1613:N = 42
1553:where
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1393:
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995:
767:
758:
602:
577:
552:
527:
456:
431:
406:
335:
310:
227:
94:
85:
1936:arXiv
1891:arXiv
1703:arXiv
1236:, or
860:exact
648:exact
1749:ISBN
1536:OEIS
1475:and
1467:and
695:(or
657:(or
646:(or
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1983:PDF
1946:doi
1901:doi
1887:181
1834:doi
1830:129
1791:doi
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1713:doi
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1379:mod
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