{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:00:32Z","timestamp":1776790832424,"version":"3.51.2"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,23]],"date-time":"2010-01-23T00:00:00Z","timestamp":1264204800000},"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 Let\n \n \n \n \n p<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n <\/mml:mrow>\n p\/q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a rational noninteger number with\n \n \n \n \n p<\/mml:mi>\n ><\/mml:mo>\n q<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n p>q\\geq 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A real number\n \n \n \n \n \n \u03bb\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\lambda >0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a\n \n \n \n \n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n Z_{p\/q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -number\n <\/italic>\n if\n \n \n \n \n {<\/mml:mo>\n \n \u03bb\n \n <\/mml:mi>\n (<\/mml:mo>\n p<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n \n )<\/mml:mo>\n n<\/mml:mi>\n <\/mml:msup>\n }<\/mml:mo>\n ><\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n <\/mml:mrow>\n \\{\\lambda (p\/q)^n\\}>1\/q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for every nonnegative integer\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n {<\/mml:mo>\n x<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n \\{x\\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes the fractional part of\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We develop several algorithms to search for\n \n \n \n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n Z_{p\/q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -numbers, and use them to determine lower bounds on such numbers for several\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n q<\/mml:mi>\n q<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . It is shown, for instance, that if there is a\n \n \n \n \n Z<\/mml:mi>\n \n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n Z_{3\/2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -number, then it is greater than\n \n \n \n \n 2<\/mml:mn>\n \n 57<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n 2^{57}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also explore some connections between these problems and some questions regarding iterated maps on integers.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02211-x","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1837-1851","source":"Crossref","is-referenced-by-count":2,"title":["Lower bounds for Z-numbers"],"prefix":"10.1090","volume":"78","author":[{"given":"Art\u016bras","family":"Dubickas","sequence":"first","affiliation":[]},{"given":"Michael","family":"Mossinghoff","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,23]]},"reference":[{"issue":"242","key":"1","doi-asserted-by":"publisher","first-page":"1035","DOI":"10.1090\/S0025-5718-02-01425-4","article-title":"Lower bounds for the total stopping time of 3\ud835\udc65+1 iterates","volume":"72","author":"Applegate, David","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8632-1","volume-title":"Pisot and Salem numbers","author":"Bertin, M.-J.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/3764326484"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"301","DOI":"10.4064\/aa114-4-1","article-title":"Linear mod one transformations and the distribution of fractional parts {\ud835\udf09(\ud835\udc5d\/\ud835\udc5e)\u207f}","volume":"114","author":"Bugeaud, Yann","year":"2004","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"4","series-title":"Progress in Physics","isbn-type":"print","volume-title":"Iterated maps on the interval as dynamical systems","volume":"1","author":"Collet, Pierre","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/3764330260"},{"issue":"4","key":"5","doi-asserted-by":"publisher","first-page":"391","DOI":"10.1007\/BF02829801","article-title":"There are infinitely many limit points of the fractional parts of powers","volume":"115","author":"Dubickas, Art\u016bras","year":"2005","journal-title":"Proc. Indian Acad. Sci. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0253-4142","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1112\/S0024609305017728","article-title":"Arithmetical properties of powers of algebraic numbers","volume":"38","author":"Dubickas, Art\u016bras","year":"2006","journal-title":"Bull. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"125","DOI":"10.4064\/aa-70-2-125-147","article-title":"On the range of fractional parts {\ud835\udf09(\ud835\udc5d\/\ud835\udc5e)\u207f}","volume":"70","author":"Flatto, Leopold","year":"1995","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1090\/conm\/135\/1185087","article-title":"\ud835\udc4d-numbers and \ud835\udefd-transformations","author":"Flatto, Leopold","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0821851462"},{"key":"9","article-title":"GMP: The GNU multiple precision arithmetic library"},{"key":"10","first-page":"250","article-title":"Ein mengentheoretischer Satz \u00fcber die Gleichverteilung modulo Eins","volume":"2","author":"Koksma, J. F.","year":"1935","journal-title":"Compositio Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-437X","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"237","DOI":"10.4064\/aa109-3-4","article-title":"Bounds for the 3\ud835\udc65+1 problem using difference inequalities","volume":"109","author":"Krasikov, Ilia","year":"2003","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1080\/10586458.2004.10504526","article-title":"Approximate squaring","volume":"13","author":"Lagarias, J. C.","year":"2004","journal-title":"Experiment. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1058-6458","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"3","DOI":"10.2307\/2322189","article-title":"The 3\ud835\udc65+1 problem and its generalizations","volume":"92","author":"Lagarias, Jeffrey C.","year":"1985","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"14","article-title":"The 3\ud835\udc65+1 Problem: An Annotated Bibliography","author":"Lagarias, J. C.","year":"2008"},{"key":"15","article-title":"The 3\ud835\udc65+1 Problem: An Annotated Bibliography, II","author":"Lagarias, J. C.","year":"2008"},{"key":"16","article-title":"Construction of a number greater than one whose powers are uniformly distributed modulo one","author":"Lerma, M. A.","year":"1996"},{"key":"17","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1017\/S1446788700005371","article-title":"An unsolved problem on the powers of 3\/2","volume":"8","author":"Mahler, K.","year":"1968","journal-title":"J. Austral. Math. Soc."},{"key":"18","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/BF02565954","article-title":"R\u00e9partition (\ud835\udc5a\ud835\udc5c\ud835\udc511) des puissances successives des nombres r\u00e9els","volume":"19","author":"Pisot, Charles","year":"1946","journal-title":"Comment. Math. Helv.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-2571","issn-type":"print"},{"issue":"1","key":"19","doi-asserted-by":"publisher","first-page":"51","DOI":"10.4064\/aa117-1-3","article-title":"Theoretical and computational bounds for \ud835\udc5a-cycles of the 3\ud835\udc5b+1-problem","volume":"117","author":"Simons, John","year":"2005","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"20","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1112\/jlms\/s1-15.2.159","article-title":"On the fractional parts of the powers of a number. I","volume":"15","author":"Vijayaraghavan, T.","year":"1940","journal-title":"J. London Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"21","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0095985","volume-title":"The dynamical system generated by the $3n+1$ function","volume":"1681","author":"Wirsching, G\u00fcnther J.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/3540639705"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02211-X\/S0025-5718-09-02211-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02211-X\/S0025-5718-09-02211-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:10:11Z","timestamp":1776787811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2009-78-267\/S0025-5718-09-02211-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,23]]},"references-count":21,"journal-issue":{"issue":"267","published-print":{"date-parts":[[2009,7]]}},"alternative-id":["S0025-5718-09-02211-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02211-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,1,23]]}}}