{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:43:16Z","timestamp":1776789796044,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,9,10]],"date-time":"2009-09-10T00:00:00Z","timestamp":1252540800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A collection of congruences with distinct moduli, each greater than\n \n \n \n 1<\/mml:mn>\n 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , such that each integer satisfies at least one of the congruences, is said to be a covering system. A famous conjecture of Erd\u00f6s from 1950 states that the least modulus of a covering system can be arbitrarily large. This conjecture remains open and, in its full strength, appears at present to be unattackable. Most of the effort in this direction has been aimed at explicitly constructing covering systems with large least modulus. Improving upon previous results of Churchhouse, Krukenberg, Choi, and Morikawa, we construct a covering system with least modulus\n \n \n \n 25<\/mml:mn>\n 25<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The construction involves a large-scale computer search, in conjunction with two general results that considerably reduce the complexity of the search.\n <\/p>","DOI":"10.1090\/s0025-5718-08-02154-6","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"1127-1146","source":"Crossref","is-referenced-by-count":3,"title":["A covering system with least modulus 25"],"prefix":"10.1090","volume":"78","author":[{"given":"Donald","family":"Gibson","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,9,10]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"73","DOI":"10.4064\/aa-48-1-73-79","article-title":"Necessary condition for the existence of an incongruent covering system with odd moduli. II","volume":"48","author":"Berger, Marc A.","year":"1987","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"885","DOI":"10.2307\/2004353","article-title":"Covering the set of integers by congruence classes of distinct moduli","volume":"25","author":"Choi, S. L. G.","year":"1971","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","first-page":"20","article-title":"Covering sets and systems of congruences","author":"Churchhouse, R. F.","year":"1968"},{"key":"4","first-page":"113","article-title":"On integers of the form 2^{\ud835\udc58}+\ud835\udc5d and some related problems","volume":"2","author":"Erd\u00f6s, P.","year":"1950","journal-title":"Summa Brasil. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0039-498X","issn-type":"print"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"74","DOI":"10.1007\/BFb0075752","article-title":"On some of my problems in number theory I would most like to see solved","author":"Erd\u0151s, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/3540152229"},{"key":"6","series-title":"Monographies de L'Enseignement Math\\'{e}matique [Monographs of L'Enseignement Math\\'{e}matique]","volume-title":"Old and new problems and results in combinatorial number theory","volume":"28","author":"Erd\u0151s, P.","year":"1980"},{"key":"7","isbn-type":"print","first-page":"1","article-title":"Coverings of the integers associated with an irreducibility theorem of A. Schinzel","author":"Filaseta, Michael","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/1568811462"},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1090\/S0894-0347-06-00549-2","article-title":"Sieving by large integers and covering systems of congruences","volume":"20","author":"Filaseta, Michael","year":"2007","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"9","unstructured":"Donald Jason Gibson, Covering systems, Ph.D. thesis, University of Illinois, Urbana-Champaign, 2006."},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.aam.2005.01.004","article-title":"On odd covering systems with distinct moduli","volume":"35","author":"Guo, Song","year":"2005","journal-title":"Adv. in Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-8858","issn-type":"print"},{"key":"11","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-26677-0","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387208607","edition":"3"},{"issue":"1","key":"12","first-page":"3","article-title":"On disjoint covering of groups by their cosets","volume":"27","author":"Korec, Ivan","year":"1977","journal-title":"Math. Slovaca","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5173","issn-type":"print"},{"key":"13","unstructured":"C. E. Krukenberg, Covering sets of the integers, Ph.D. thesis, University of Illinois, Urbana-Champaign, 1971."},{"issue":"1","key":"14","first-page":"1","article-title":"On a method to construct covering sets","volume":"22","author":"Morikawa, Ryozo","year":"1981","journal-title":"Bull. Fac. Liberal Arts Nagasaki Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0287-1300","issn-type":"print"},{"issue":"2","key":"15","first-page":"1","article-title":"Some examples of covering sets","volume":"21","author":"Morikawa, Ryozo","year":"1981","journal-title":"Bull. Fac. Liberal Arts Nagasaki Univ.","ISSN":"https:\/\/id.crossref.org\/issn\/0287-1300","issn-type":"print"},{"key":"16","unstructured":"Pace Nielsen, A covering system whose smallest modulus is large, preprint."},{"key":"17","isbn-type":"print","first-page":"581","article-title":"Covering systems of Paul Erd\u0151s. Past, present and future","author":"Porubsk\u00fd, \u0160.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/3540422366"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"668","DOI":"10.1007\/BF01449161","article-title":"\u00dcber einige S\u00e4tze der additiven Zahlentheorie","volume":"109","author":"Romanoff, N. P.","year":"1934","journal-title":"Math. Ann.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5831","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"91","DOI":"10.4064\/aa-13-1-91-101","article-title":"Reducibility of polynomials and covering systems of congruences","volume":"13","author":"Schinzel, A.","year":"1967","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"20","first-page":"21","article-title":"Covering congruences related to modular arithmetic and error correcting codes","volume":"16","author":"Sch\u00f6nheim, J.","year":"1983","journal-title":"Ars Combin.","ISSN":"https:\/\/id.crossref.org\/issn\/0381-7032","issn-type":"print"},{"key":"21","first-page":"73","article-title":"Sur un probl\u00e8me concernant les nombres \ud835\udc58\u22c52\u207f+1","volume":"15","author":"Sierpi\u0144ski, W.","year":"1960","journal-title":"Elem. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0013-6018","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"59","DOI":"10.4064\/aa-59-1-59-70","article-title":"Necessary conditions for distinct covering systems with square-free moduli","volume":"59","author":"Simpson, R. J.","year":"1991","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"23","first-page":"1","article-title":"Word problems requiring exponential time: preliminary report","author":"Stockmeyer, L. J.","year":"1973"},{"key":"24","unstructured":"J. D. Swift, Sets of covering congruences, Bull. Amer. Math. Soc. 60 (1954), 390."},{"key":"25","first-page":"59","article-title":"A survey of covering systems of congruences","volume":"40\/41","author":"Zn\u00e1m, \u0160tefan","year":"1982","journal-title":"Acta Math. Univ. 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