{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:51:05Z","timestamp":1776844265236,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,10,4]],"date-time":"2002-10-04T00:00:00Z","timestamp":1033689600000},"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 Erd\u0151s conjectured that there are\n \n \n \n \n x<\/mml:mi>\n \n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n x^{1-o(1)}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n Carmichael numbers up to\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , whereas Shanks was skeptical as to whether one might even find an\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n up to which there are more than\n \n \n \n \n x<\/mml:mi>\n <\/mml:msqrt>\n \\sqrt {x}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n Carmichael numbers. Alford, Granville and Pomerance showed that there are more than\n \n \n \n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 7<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n x^{2\/7}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n Carmichael numbers up to\n \n \n \n x<\/mml:mi>\n x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and gave arguments which even convinced Shanks (in person-to-person discussions) that Erd\u0151s must be correct. Nonetheless, Shanks\u2019s skepticism stemmed from an appropriate analysis of the data available to him (and his reasoning is still borne out by Pinch\u2019s extended new data), and so we herein derive conjectures that are consistent with Shanks\u2019s observations, while fitting in with the viewpoint of Erd\u0151s and the results of Alford, Granville and Pomerance.\n <\/p>","DOI":"10.1090\/s0025-5718-01-01355-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"883-908","source":"Crossref","is-referenced-by-count":21,"title":["Two contradictory conjectures concerning Carmichael numbers"],"prefix":"10.1090","volume":"71","author":[{"given":"Andrew","family":"Granville","sequence":"first","affiliation":[]},{"given":"Carl","family":"Pomerance","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,10,4]]},"reference":[{"key":"1","unstructured":"[1] W. R. Alford and J. Grantham, Carmichael numbers with exactly \ud835\udc58 prime factors (to appear)."},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"703","DOI":"10.2307\/2118576","article-title":"There are infinitely many Carmichael numbers","volume":"139","author":"Alford, W. R.","year":"1994","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/3-540-58691-1_36","article-title":"On the difficulty of finding reliable witnesses","author":"Alford, W. R.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/3540586911"},{"issue":"220","key":"4","doi-asserted-by":"publisher","first-page":"1705","DOI":"10.1090\/S0025-5718-97-00857-0","article-title":"Density of Carmichael numbers with three prime factors","volume":"66","author":"Balasubramanian, R.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-314X(83)90002-1","article-title":"On a problem of Oppenheim concerning \u201cfactorisatio numerorum\u201d","volume":"17","author":"Canfield, E. R.","year":"1983","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"issue":"203","key":"6","doi-asserted-by":"publisher","first-page":"177","DOI":"10.2307\/2152945","article-title":"Average case error estimates for the strong probable prime test","volume":"61","author":"Damg\u00e5rd, Ivan","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/crll.1940.181.1","article-title":"Untersuchungen \u00fcber reinverzweigte Erweiterungen diskret bewerteter perfekter K\u00f6rper","volume":"181","author":"Arf, Cahit","year":"1939","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"8","volume-title":"New First Course in the Theory of Equations","author":"Dickson, Leonard Eugene","year":"1939"},{"key":"9","unstructured":"[10] W. Galway, The density of pseudoprimes with two prime factors (to appear)."},{"key":"10","unstructured":"[11] A. Granville, Primality testing and Carmichael numbers, Notices Amer. Math. Soc. 39 (1992), 696\u2013700."},{"key":"11","doi-asserted-by":"crossref","unstructured":"[12] G. H. Hardy and J. E. Littlewood, Some problems on partitio numerorum III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1\u201370.","DOI":"10.1007\/BF02403921"},{"issue":"148","key":"12","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1093\/qmath\/37.4.401","article-title":"Local densities over integers free of large prime factors","volume":"37","author":"Ivi\u0107, Aleksandar","year":"1986","journal-title":"Quart. J. Math. Oxford Ser. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"issue":"2","key":"13","first-page":"13","article-title":"Galois fields and computation","author":"Li, Yong","year":"1991","journal-title":"Neimenggu Shida Xuebao Ziran Kexue Ban"},{"key":"14","unstructured":"[15] R. G. E. Pinch, The Carmichael numbers up to 10\u00b9\u2076 (to appear)."},{"key":"15","unstructured":"[16] R. G. E. Pinch, The pseudoprimes up to 10\u00b9\u00b2 (to appear)."},{"issue":"156","key":"16","doi-asserted-by":"publisher","first-page":"587","DOI":"10.2307\/2007448","article-title":"On the distribution of pseudoprimes","volume":"37","author":"Pomerance, Carl","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"17","unstructured":"[19] C. Pomerance, Two methods in elementary analytic number theory, Number Theory and Applications (Banff, 1988; R. A. Mollin, ed.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 265, Reidel, Dordrecht, 1989, pp. 135\u2013161."},{"issue":"3","key":"18","first-page":"199","article-title":"Carmichael numbers","volume":"11","author":"Pomerance, Carl","year":"1993","journal-title":"Nieuw Arch. Wisk. (4)","ISSN":"https:\/\/id.crossref.org\/issn\/0028-9825","issn-type":"print"},{"issue":"151","key":"19","doi-asserted-by":"publisher","first-page":"1003","DOI":"10.2307\/2006210","article-title":"The pseudoprimes to 25\u22c510\u2079","volume":"35","author":"Pomerance, Carl","year":"1980","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"20","doi-asserted-by":"publisher","first-page":"185","DOI":"10.4064\/aa-4-3-185-208","article-title":"Sur certaines hypoth\u00e8ses concernant les nombres premiers","volume":"4","author":"Schinzel, A.","year":"1958","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1215\/S0012-7094-39-00546-6","article-title":"Certain congruences involving the Bernoulli numbers","volume":"5","author":"Vandiver, H. S.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"key":"22","isbn-type":"print","volume-title":"Solved and unsolved problems in number theory","author":"Shanks, Daniel","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0828412979","edition":"3"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01355-2\/S0025-5718-01-01355-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01355-2\/S0025-5718-01-01355-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:57:01Z","timestamp":1776725821000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01355-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,10,4]]},"references-count":22,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01355-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01355-2","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":[[2001,10,4]]}}}