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index modular arithmetic Carmichael numbers conjecture |
| It is conjectured that every odd number coprime to its Euler totient divides some Carmichael number. Only numbers satisfying this necessary condition are listed here; a question mark indicates those for which the Carmichael multiple given need not be the least one: 885, 2391, 2571, 2589, 2595, 2685, 2949. The list on this page dates back to 2003 (see footnotes). It was updated in 2007, shortly before we published (jointly with Joe Crump) a similar list extended to all divisors below 10000. See also our list of Carmichael multiples of primes up to 10000. |
1: 561 = 3.11.17 3: 561 = 3.11.17 5: 1105 = 5.13.17 7: 1729 = 7.13.19 11: 561 = 3.11.17 13: 1105 = 5.13.17 15: 62745 = 3.5.47.89 17: 561 = 3.11.17 19: 1729 = 7.13.19 23: 6601 = 7.23.41 29: 2465 = 5.17.29 31: 2821 = 7.13.31 33: 561 = 3.11.17 35: 825265 = 5.7.17.19.73 37: 29341 = 13.37.61 41: 6601 = 7.23.41 43: 334153 = 19.43.409 47: 62745 = 3.5.47.89 51: 561 = 3.11.17 53: 2433601 = 17.37.53.73 59: 74165065 = 5.13.59.83.233 61: 29341 = 13.37.61 65: 1105 = 5.13.17 67: 8911 = 7.19.67 69: 116150434401 = 3.23.101.2081.8009 71: 10024561 = 71.271.521 73: 10585 = 5.29.73 77: 41041 = 7.11.13.41 79: 2508013 = 53.79.599 83: 55462177 = 17.23.83.1709 85: 1105 = 5.13.17 87: 11921001 = 3.29.263.521 89: 62745 = 3.5.47.89 91: 1729 = 7.13.19 95: 449065 = 5.19.29.163 97: 46657 = 13.37.97 101: 101101 = 7.11.13.101 103: 52633 = 7.73.103 107: 84350561 = 107.743.1061 109: 188461 = 7.13.19.109 113: 278545 = 5.17.29.113 115: 1050985 = 5.13.19.23.37 119: 825265 = 5.7.17.19.73 123: 689880801 = 3.41.71.197.401 127: 1152271 = 43.127.211 131: 18307381 = 29.61.79.131 133: 1729 = 7.13.19 137: 410041 = 41.73.137 139: 2628073 = 7.37.73.139 141: 62745 = 3.5.47.89 143: 41041 = 7.11.13.41 145: 2465 = 5.17.29 149: 12261061 = 19.61.71.149 151: 838201 = 7.13.61.151 157: 4335241 = 53.157.521 159: 230996949 = 3.53.317.4583 161: 6601 = 7.23.41 163: 449065 = 5.19.29.163 167: 18633869185 = 5.43.167.193.2689 173: 45318561 = 3.29.173.3011 177: 939161538561 = 3.59.233.281.81041 179: 851703301 = 179.1069.4451 181: 488881 = 37.73.181 185: 1050985 = 5.13.19.23.37 187: 561 = 3.11.17 191: 34657141 = 191.421.431 193: 3224065 = 5.13.193.257 197: 656601 = 3.11.101.197 199: 670033 = 7.13.37.199 209: 6313681 = 11.17.19.1777 211: 399001 = 31.61.211 213: 174352641 = 3.71.641.1277 215: 115542505 = 5.13.43.67.617 217: 2821 = 7.13.31 221: 1105 = 5.13.17 223: 977892241 = 17.31.53.157.223 227: 2436691321 = 227.1583.6781 229: 6840001 = 7.17.229.251 233: 162401 = 17.41.233 235: 62745 = 3.5.47.89 239: 9613297 = 19.29.73.239 241: 115921 = 13.37.241 247: 1729 = 7.13.19 249: 1070730898161 = 3.29.83.89.239.6971 251: 3828001 = 101.151.251 255: 1886616373665 = 3.5.17.23.83.353.10979 257: 3224065 = 5.13.193.257 259: 63973 = 7.13.19.37 263: 11921001 = 3.29.263.521 265: 6054985 = 5.53.73.313 267: 62745 = 3.5.47.89 269: 962500561 = 11.239.269.1361 271: 512461 = 31.61.271 277: 35703361 = 61.277.2113 281: 29111881 = 211.281.491 283: 182356993 = 7.13.73.97.283 287: 6601 = 7.23.41 293: 790623289 = 37.233.293.313 295: 74165065 = 5.13.59.83.233 299: 340561 = 13.17.23.67 303: 656601 = 3.11.101.197 307: 1033669 = 7.13.37.307 311: 181154701 = 7.13.37.173.311 313: 6054985 = 5.53.73.313 317: 230996949 = 3.53.317.4583 319: 2704801 = 11.29.61.139 321: 17544679041 = 3.41.107.449.2969 323: 825265 = 5.7.17.19.73 329: 2574243721 = 7.47.139.181.311 331: 1857241 = 31.181.331 335: 40280065 = 5.7.67.89.193 337: 3057601 = 43.211.337 339: 42887804961 = 3.71.89.113.20021 341: 75361 = 11.13.17.31 345: 1886616373665 = 3.5.17.23.83.353.10979 347: 34087042891 = 11.19.307.347.1531 349: 413058601 = 31.73.349.523 353: 1615681 = 23.199.353 359: 325546585 = 5.7.13.359.1993 365: 10585 = 5.29.73 367: 257495641 = 13.31.367.1741 371: 13992265 = 5.7.19.53.397 373: 46483633 = 7.19.373.937 377: 10402561 = 13.29.41.673 379: 6733693 = 109.163.379 383: 1678569121 = 31.37.383.3821 389: 1955324449 = 13.29.67.199.389 391: 340561 = 13.17.23.67 393: 1848112761 = 3.11.41.131.10427 395: 1074363265 = 5.19.37.53.73.79 397: 314821 = 13.61.397 401: 8341201 = 11.31.61.401 403: 2821 = 7.13.31 407: 29020321 = 11.37.113.631 409: 334153 = 19.43.409 411: 58856010081 = 3.11.29.41.137.10949 413: 544101481 = 7.11.59.229.523 415: 74165065 = 5.13.59.83.233 419: 426821473 = 13.127.419.617 421: 530881 = 13.97.421 427: 172081 = 7.13.31.61 431: 34657141 = 191.421.431 433: 748657 = 7.13.19.433 435: 62267183533185 = 3.5.29.449.3137.101627 437: 1050985 = 5.13.19.23.37 439: 4382684065 = 5.17.67.439.1753 443: 60112885 = 5.7.443.3877 445: 62745 = 3.5.47.89 447: 4173049274241 = 3.23.149.281.1444481 449: 5632705 = 5.13.193.449 451: 41041 = 7.11.13.41 455: 41298985 = 5.7.13.139.653 457: 16046641 = 13.37.73.457 461: 1909001 = 41.101.461 463: 10267951 = 67.331.463 467: 14005499521 = 13.17.97.467.1399 469: 8911 = 7.19.67 473: 10877581 = 11.13.29.43.61 479: 1441316269 = 19.29.43.127.479 481: 29341 = 13.37.61 485: 9582145 = 5.23.97.859 487: 64377991 = 163.487.811 491: 14676481 = 71.421.491 493: 2465 = 5.17.29 499: 2230305949 = 499.997.4483 501: 236359158267 = 3.47.167.239.41999 503: 175997185 = 5.7.13.503.769 509: 11119105 = 5.17.257.509 511: 15841 = 7.31.73 515: 130032865 = 5.19.97.103.137 517: 26932081 = 11.47.113.461 519: 45318561 = 3.29.173.3011 521: 4335241 = 53.157.521 523: 14469841 = 73.379.523 527: 75361 = 11.13.17.31 533: 41041 = 7.11.13.41 535: 413631505 = 5.7.17.73.89.107 537: 675961912209921 = 3.17.179.281.13217.19937 541: 1461241 = 37.73.541 545: 67371265 = 5.13.37.109.257 547: 15829633 = 43.547.673 551: 449065 = 5.19.29.163 553: 72286501 = 7.67.79.1951 557: 8270944801 = 7.31.41.557.1669 559: 10877581 = 11.13.29.43.61 561: 561 = 3.11.17 563: 80414269057 = 17.67.89.563.1409 565: 278545 = 5.17.29.113 569: 6879465481 = 7.41.103.409.569 571: 9439201 = 61.271.571 573: 304044043521 = 3.191.257.641.3221 577: 997633 = 7.13.19.577 581: 3034203361 = 7.83.821.6361 583: 173085121 = 11.31.53.61.157 587: 114919915021 = 587.5861.33403 589: 2113921 = 19.31.37.97 591: 656601 = 3.11.101.197 593: 28787185 = 5.7.19.73.593 595: 825265 = 5.7.17.19.73 599: 2508013 = 53.79.599 601: 5049001 = 31.271.601 607: 94536001 = 7.19.607.1171 611: 945959365 = 5.13.19.43.47.379 613: 36765901 = 37.613.1621 617: 19384289 = 89.353.617 619: 38637361 = 7.37.241.619 623: 40280065 = 5.7.67.89.193 629: 2433601 = 17.37.53.73 631: 1193221 = 31.61.631 635: 3664585 = 5.29.127.199 641: 70561921 = 31.53.67.641 643: 11972017 = 43.433.643 647: 90698401 = 103.647.1361 649: 544101481 = 7.11.59.229.523 653: 41298985 = 5.7.13.139.653 659: 114910489 = 127.659.1373 661: 34901461 = 7.19.397.661 665: 825265 = 5.7.17.19.73 667: 5031181 = 19.23.29.397 671: 852841 = 11.31.41.61 673: 10402561 = 13.29.41.673 677: 2642025673 = 7.53.67.157.677 679: 182356993 = 7.13.73.97.283 681: 77403893505 = 3.5.197.227.257.449 683: 84311569 = 19.73.89.683 685: 130032865 = 5.19.97.103.137 691: 79624621 = 139.691.829 695: 15403285 = 5.37.139.599 697: 162401 = 17.41.233 699: 939161538561 = 3.59.233.281.81041 701: 31405501 = 71.631.701 703: 63973 = 7.13.19.37 705: 62745 = 3.5.47.89 707: 101101 = 7.11.13.101 709: 547652161 = 7.17.709.6491 713: 683379841 = 23.31.41.97.241 717: 101536702401 = 3.197.239.449.1601 719: 110765400481 = 17.719.757.11971 721: 52633 = 7.73.103 727: 278152381 = 13.19.727.1549 731: 1569457 = 17.19.43.113 733: 295643089 = 7.157.367.733 739: 4909177 = 7.13.73.739 743: 84350561 = 107.743.1061 745: 745864945 = 5.7.89.149.1607 749: 413631505 = 5.7.17.73.89.107 751: 68154001 = 151.601.751 753: 7281824001 = 3.251.2417.4001 757: 6868261 = 43.211.757 761: 7519441 = 41.241.761 763: 188461 = 7.13.19.109 767: 74165065 = 5.13.59.83.233 769: 50201089 = 97.673.769 771: 77403893505 = 3.5.197.227.257.449 773: 419520241 = 7.31.41.61.773 779: 16778881 = 7.17.19.41.181 781: 67994641 = 11.13.37.71.181 785: 31692805 = 5.47.157.859 787: 542497201 = 263.787.2621 789: 11921001 = 3.29.263.521 793: 29341 = 13.37.61 795: 294628486347105 = 3.5.53.113.11699.280337 797: 5685601141 = 11.181.797.3583 799: 20964961 = 17.19.47.1381 803: 4903921 = 11.31.73.197 805: 225745345 = 5.7.23.193.1453 807: 60580578044001 = 3.17.269.20369.216791 809: 44378446561 = 809.5657.9697 811: 64377991 = 163.487.811 815: 449065 = 5.19.29.163 817: 334153 = 19.43.409 821: 2438403661 = 739.821.4019 823: 8346731851 = 823.2467.4111 827: 1746692641 = 113.827.18691 829: 79624621 = 139.691.829 835: 18633869185 = 5.43.167.193.2689 839: 330983586241 = 7.11.839.1553.3299 843: 6097778961 = 3.131.281.55217 851: 1050985 = 5.13.19.23.37 853: 7995169 = 7.13.103.853 857: 148689562561 = 71.673.857.3631 859: 9582145 = 5.23.97.859 863: 255160621 = 37.61.131.863 865: 92429188345 = 5.13.73.109.173.1033 869: 105869401 = 11.29.79.4201 871: 340561 = 13.17.23.67 877: 1200778753 = 293.877.4673 879: 836229561461841 = 3.53.113.293.509.521.599 881: 128697361 = 13.17.661.881 883: 1349671681 = 37.109.379.883 885: 24354644805191195265 = 3.5.59.449.1373.205553.217169 [?] 887: 981567505 = 5.17.47.277.887 893: 20964961 = 17.19.47.1381 895: 75150127585 = 5.13.179.1303.4957 899: 765245881 = 19.29.31.71.631 901: 2433601 = 17.37.53.73 907: 18900973 = 7.13.229.907 911: 40622401 = 17.43.61.911 913: 4375257601 = 11.13.83.449.821 917: 393716701 = 7.53.131.8101 919: 2531845 = 5.19.29.919 923: 67994641 = 11.13.37.71.181 929: 600892993 = 19.59.577.929 933: 7357194781706001 = 3.41.89.113.311.701.27281 937: 46483633 = 7.19.373.937 941: 507726901 = 11.181.271.941 943: 6601 = 7.23.41 947: 7558388641 = 331.947.24113 949: 126217 = 7.13.19.73 951: 230996949 = 3.53.317.4583 953: 5444489 = 29.197.953 957: 1534274841 = 3.11.29.479.3347 959: 302751505 = 5.7.103.137.613 965: 3224065 = 5.13.193.257 967: 53245921 = 17.41.79.967 971: 6155121421 = 19.37.71.127.971 973: 2628073 = 7.37.73.139 977: 3222053185 = 5.13.113.449.977 983: 34138047673 = 983.3929.8839 985: 21584305 = 5.17.197.1289 989: 111291181 = 23.43.131.859 991: 8830801 = 7.19.67.991 995: 3664585 = 5.29.127.199 997: 1950276565 = 5.19.59.349.997 1001: 41041 = 7.11.13.41 1003: 2709611521 = 17.59.61.67.661 1007: 13992265 = 5.7.19.53.397 1009: 84417985 = 5.29.577.1009 1013: 413138881 = 617.661.1013 1019: 12760344961 = 7.13.23.31.193.1019 1021: 473847121 = 11.31.1021.1361 1031: 434330401 = 43.97.101.1031 1033: 4210922233 = 7.37.1033.15739 1037: 552721 = 13.17.41.61 1039: 130497361 = 29.61.71.1039 1041: 3776882591682201 = 3.53.89.137.347.701.8009 1043: 745864945 = 5.7.89.149.1607 1049: 939947009 = 263.1049.3407 1051: 58489201 = 19.29.101.1051 1057: 838201 = 7.13.61.151 1059: 28173681537 = 3.353.2699.9857 1061: 84350561 = 107.743.1061 1063: 3102234751 = 7.11.151.251.1063 1067: 172430401 = 11.13.31.97.401 1069: 851703301 = 179.1069.4451 1073: 12945745 = 5.19.29.37.127 1077: 480656935630257 = 3.197.359.419.1163.4649 1079: 74165065 = 5.13.59.83.233 1087: 18152272801 = 13.17.19.41.97.1087 1091: 3347570941 = 11.13.43.499.1091 1093: 3581761 = 29.113.1093 1097: 13657063105 = 5.7.19.97.193.1097 1099: 43620409 = 7.19.157.2089 1103: 29465082049 = 13.269.1103.7639 1105: 1105 = 5.13.17 1109: 8885251441 = 11.47.1109.15497 1111: 101101 = 7.11.13.101 1115: 10967410945 = 5.19.223.449.1153 1117: 775368901 = 373.1117.1861 1121: 338740417 = 19.59.449.673 1123: 743404663 = 103.1123.6427 1129: 529782121 = 11.29.1129.1471 1133: 557795161 = 7.11.53.103.1327 1135: 77403893505 = 3.5.197.227.257.449 1139: 340561 = 13.17.23.67 1141: 1082809 = 7.13.73.163 1145: 231194965 = 5.23.229.8779 1147: 2113921 = 19.31.37.97 1149: 5949649907586561 = 3.11.83.257.383.1373.16073 1151: 104569501 = 47.1151.1933 1153: 176659201 = 37.41.101.1153 1157: 226509361 = 13.89.137.1429 1159: 2455921 = 13.19.61.163 1163: 56549585233 = 1163.4649.10459 1165: 74165065 = 5.13.59.83.233 1167: 655510549443465 = 3.5.23.53.389.2663.34607 1169: 18660149641 = 7.19.31.41.167.661 1171: 5481451 = 31.151.1171 1173: 1886616373665 = 3.5.17.23.83.353.10979 1177: 4752717761 = 11.17.107.173.1373 1181: 193708801 = 11.13.31.37.1181 1187: 27968406181 = 31.53.1187.14341 1189: 10402561 = 13.29.41.673 1193: 957044881 = 13.23.1193.2683 1195: 1999004365 = 5.29.37.239.1559 1199: 100427041 = 11.13.17.109.379 1201: 7207201 = 17.353.1201 1203: 684106401 = 3.11.17.401.3041 1207: 75151441 = 17.19.29.71.113 1211: 181154701 = 7.13.37.173.311 1213: 6693621481 = 607.1213.9091 1217: 1337805505 = 5.109.1217.2017 1219: 2140538401 = 23.29.53.151.401 1223: 15039215101 = 31.37.71.151.1223 1229: 10550902321 = 17.41.109.113.1229 1231: 897880321 = 13.19.1231.2953 1235: 1050985 = 5.13.19.23.37 1237: 1420379065 = 5.7.53.619.1237 1241: 825265 = 5.7.17.19.73 1243: 6049681 = 11.31.113.157 1245: 1886616373665 = 3.5.17.23.83.353.10979 1247: 10877581 = 11.13.29.43.61 1249: 23382529 = 97.193.1249 1253: 1362132541 = 7.11.37.179.2671 1257: 5860426881 = 3.17.41.419.6689 1259: 106339394161 = 7.11.13.19.1259.4441 1261: 46657 = 13.37.97 1267: 4463641 = 7.13.181.271 1271: 852841 = 11.31.41.61 1273: 8911 = 7.19.67 1277: 174352641 = 3.71.641.1277 1279: 1194866101 = 11.13.47.139.1279 1283: 63280622521 = 1283.3847.12821 1285: 3224065 = 5.13.193.257 1289: 21584305 = 5.17.197.1289 1291: 517937581 = 7.37.1291.1549 1293: 199438988001 = 3.11.41.71.431.4817 1295: 1615335085 = 5.7.13.37.229.419 1297: 171454321 = 163.811.1297 1301: 27062101 = 11.31.61.1301 1303: 955134181 = 311.1303.2357 1307: 2111488561 = 7.13.41.433.1307 1309: 2313774001 = 7.11.17.19.31.3001 1313: 101101 = 7.11.13.101 1315: 959694999145 = 5.29.47.263.277.1933 1319: 144377609419 = 1319.9227.11863 1321: 8719921 = 7.23.41.1321 1327: 557795161 = 7.11.53.103.1327 1329: 55086689841 = 3.29.41.71.443.491 1333: 8927101 = 31.37.43.181 1335: 62745 = 3.5.47.89 1337: 72108421 = 7.11.191.4903 1339: 7995169 = 7.13.103.853 1343: 53245921 = 17.41.79.967 1345: 12782849065 = 5.7.269.317.4283 1347: 10031651841 = 3.29.71.449.3617 1349: 12261061 = 19.61.71.149 1351: 37167361 = 7.11.41.61.193 1353: 1848112761 = 3.11.41.131.10427 1357: 3284630713 = 7.13.23.59.67.397 1361: 90698401 = 103.647.1361 1363: 13402361281 = 17.29.47.67.89.97 1367: 157596376201 = 11.31.233.1367.1451 1373: 13187665 = 5.17.113.1373 1381: 15247621 = 61.181.1381 1383: 180481509681 = 3.11.17.41.461.17021 1385: 334783585 = 5.37.47.139.277 1387: 126217 = 7.13.19.73 1391: 4521794641 = 7.13.23.61.107.331 1393: 670033 = 7.13.37.199 1397: 631071001 = 11.29.37.127.421 1399: 14005499521 = 13.17.97.467.1399 1401: 2220232577722881 = 3.197.449.467.1373.13049 1403: 9494101 = 23.61.67.101 1409: 5447713921 = 13.23.67.193.1409 1411: 55462177 = 17.23.83.1709 1415: 1332521065 = 5.283.659.1429 1417: 188461 = 7.13.19.109 1423: 1442761201 = 7.241.601.1423 1427: 4310730293521 = 311.373.1427.26041 1429: 226509361 = 13.89.137.1429 1433: 88183003921 = 1433.4297.14321 1437: 1534274841 = 3.11.29.479.3347 1439: 2803378414477 = 1439.8629.225767 1441: 697906561 = 11.97.131.4993 1447: 35571601 = 13.31.61.1447 1451: 153589801 = 151.701.1451 1453: 225745345 = 5.7.23.193.1453 1457: 237597361 = 31.47.313.521 1459: 129762001 = 19.31.151.1459 1463: 1419339691 = 7.11.19.103.9419 1465: 28021044745 = 5.109.293.379.463 1469: 25603201 = 13.29.113.601 1471: 529782121 = 11.29.1129.1471 1473: 7397902401 = 3.491.1601.3137 1479: 2176838049 = 3.17.29.317.4643 1481: 271794601 = 13.19.743.1481 1483: 18929704561 = 17.31.53.457.1483 1487: 77111274241 = 37.79.113.157.1487 1489: 1309440001 = 41.89.241.1489 1493: 502565593321 = 11.1493.2857.10711 1495: 1050985 = 5.13.19.23.37 1499: 66828386521 = 13.211.1499.16253 1501: 8719309 = 19.37.79.157 1507: 77826001 = 11.43.137.1201 1509: 881715504450705 = 3.5.17.47.89.113.503.14543 1511: 1521835381 = 11.19.61.79.1511 1513: 121247281 = 17.89.127.631 1517: 48321001 = 37.41.53.601 1523: 13799091241 = 13.41.89.191.1523 1527: 53868563841 = 3.53.131.509.5081 1529: 2704801 = 11.29.61.139 1531: 288120421 = 307.613.1531 1535: 602074585 = 5.7.137.307.409 1537: 45890209 = 29.53.73.409 1541: 340561 = 13.17.23.67 1543: 371345426881 = 11.13.853.1543.1973 1547: 2560104001 = 7.13.17.41.181.223 1549: 278152381 = 13.19.727.1549 1551: 126394016001 = 3.11.47.101.701.1151 1553: 1672719217 = 13.29.1553.2857 1559: 26719701 = 3.29.197.1559 1561: 2199700321 = 7.13.61.223.1777 1563: 11921001 = 3.29.263.521 1565: 6054985 = 5.53.73.313 1567: 2201169601 = 31.113.401.1567 1571: 1301203515361 = 1571.25121.32971 1577: 18785465281 = 11.19.31.83.181.193 1579: 41812581601 = 7.37.1579.102241 1583: 2436691321 = 227.1583.6781 1585: 1776450565 = 5.317.523.2143 1589: 18382122451 = 7.11.19.227.55351 1591: 8927101 = 31.37.43.181 1597: 43331401 = 43.631.1597 1601: 33302401 = 11.31.61.1601 1603: 6840001 = 7.17.229.251 1605: 607368553651905 = 3.5.107.947.15137.26399 1607: 745864945 = 5.7.89.149.1607 1609: 11089049401 = 19.31.1609.11701 1613: 33300199505 = 5.17.89.1613.2729 1615: 825265 = 5.7.17.19.73 1619: 666676060161 = 3.17.89.257.353.1619 1621: 36765901 = 37.613.1621 1627: 60246018673 = 1627.3253.11383 1631: 2339165521 = 7.11.233.241.541 1633: 836515681 = 23.71.97.5281 1637: 153033820381 = 13.19.29.31.421.1637 1639: 9439491061 = 7.11.13.19.149.3331 1643: 70561921 = 31.53.67.641 1645: 6295936465 = 5.7.13.37.47.73.109 1649: 22665505 = 5.17.97.2749 1651: 426821473 = 13.127.419.617 1657: 227752993 = 13.97.109.1657 1661: 40430401 = 11.101.151.241 1663: 13330320301 = 61.331.397.1663 1667: 569332177 = 239.1429.1667 1669: 8270944801 = 7.31.41.557.1669 1671: 3247889259130881 = 3.239.557.641.953.13313 1679: 934784929 = 13.23.73.113.379 1685: 36121345 = 5.13.17.97.337 1687: 9890881 = 7.11.13.41.241 1689: 158353658932305 = 3.5.17.89.149.563.83177 1691: 84311569 = 19.73.89.683 1693: 10779325921 = 7.13.31.37.61.1693 1695: 294628486347105 = 3.5.53.113.11699.280337 1697: 3249390145 = 5.113.1697.3389 1699: 11542494601 = 11.43.53.271.1699 1707: 1334060626449 = 3.17.89.569.516539 1709: 55462177 = 17.23.83.1709 1717: 151530401 = 11.17.71.101.113 1721: 153927961 = 11.47.173.1721 1723: 5165559169 = 43.113.617.1723 1727: 6049681 = 11.31.113.157 1729: 1729 = 7.13.19 1733: 155999871721 = 1733.5197.17321 1735: 1238761176445 = 5.13.19.347.523.5527 1739: 334783585 = 5.37.47.139.277 1741: 257495641 = 13.31.367.1741 1745: 1950276565 = 5.19.59.349.997 1747: 4212413569 = 7.13.1747.26497 1749: 2249558743641 = 3.11.53.59.4421.4931 1753: 393122521 = 11.19.29.37.1753 1757: 6840001 = 7.17.229.251 1759: 97887454669 = 1759.3517.15823 1761: 4449488694321 = 3.11.587.5861.39191 1763: 83966401 = 41.43.97.491 1765: 429553345 = 5.13.97.193.353 1769: 2704801 = 11.29.61.139 1777: 6313681 = 11.17.19.1777 1779: 6243651372801 = 3.59.593.641.92801 1781: 145124785 = 5.13.43.137.379 1783: 15888313 = 7.19.67.1783 1787: 1527864562081 = 11.13.37.283.571.1787 1789: 3441837421 = 11.29.37.163.1789 1793: 2797002901 = 7.11.19.37.163.317 1795: 325546585 = 5.7.13.359.1993 1797: 718345813641 = 3.41.53.71.599.2591 1799: 1210178305 = 5.7.19.73.97.257 1801: 48628801 = 13.31.67.1801 1807: 41298985 = 5.7.13.139.653 1811: 1163659861 = 7.13.23.307.1811 1817: 106041937 = 17.23.79.3433 1819: 413631505 = 5.7.17.73.89.107 1823: 3695285036521 = 13.29.631.1823.8521 1829: 638959321 = 7.11.13.31.59.349 1831: 492559141 = 367.733.1831 1835: 99367251985 = 5.29.37.109.367.463 1837: 288572076001 = 11.73.151.167.14251 1841: 83056054081 = 7.11.13.31.263.10177 1843: 2113921 = 19.31.37.97 1847: 660950414671 = 1847.12923.27691 1851: 662086041 = 3.89.617.4019 1853: 100427041 = 11.13.17.109.379 1855: 13992265 = 5.7.19.53.397 1861: 775368901 = 373.1117.1861 1865: 41135881645 = 5.37.157.373.3797 1867: 663162938677 = 1867.11197.31723 1871: 139952671 = 131.571.1871 1873: 1027334881 = 53.79.131.1873 1877: 63734663077 = 269.1877.126229 1879: 835085057941 = 1879.16903.26293 1883: 12782849065 = 5.7.269.317.4283 1885: 440707345 = 5.13.29.113.2069 1889: 99861985 = 5.97.109.1889 1891: 172081 = 7.13.31.61 1895: 145124785 = 5.13.43.137.379 1897: 4463641 = 7.13.181.271 1901: 2690867401 = 191.1901.7411 1903: 153927961 = 11.47.173.1721 1907: 97225715665 = 5.13.17.29.37.43.1907 1909: 55462177 = 17.23.83.1709 1913: 83946864769 = 1913.5737.7649 1915: 55085381935585 = 5.23.383.8009.156157 1919: 58489201 = 19.29.101.1051 1921: 278545 = 5.17.29.113 1923: 174352641 = 3.71.641.1277 1927: 9216037441 = 13.37.41.47.61.163 1931: 238394136541 = 11.43.211.1237.1931 1933: 11205601 = 11.17.31.1933 1937: 616463809 = 13.17.97.149.193 1939: 271481329 = 7.19.277.7369 1941: 7013994612801 = 3.11.281.491.647.2381 1943: 134857801 = 13.19.29.67.281 1945: 94706509885 = 5.13.197.389.19013 1947: 2249558743641 = 3.11.53.59.4421.4931 1949: 482368435201 = 19.101.1949.128971 1951: 72286501 = 7.67.79.1951 1955: 23839510465 = 5.17.23.29.199.2113 1957: 80927821 = 13.19.103.3181 1959: 39509680805445 = 3.5.653.27059.149069 1961: 2433601 = 17.37.53.73 1963: 838201 = 7.13.61.151 1969: 1362132541 = 7.11.37.179.2671 1973: 94572608065 = 5.13.349.1973.2113 1977: 1678722910593 = 3.29.53.113.659.4889 1979: 2117725921 = 17.19.1979.3313 1981: 182356993 = 7.13.73.97.283 1985: 13992265 = 5.7.19.53.397 1987: 14983655041 = 17.31.41.349.1987 1991: 67994641 = 11.13.37.71.181 1993: 325546585 = 5.7.13.359.1993 1997: 650186321401 = 41.73.181.601.1997 1999: 31956012001 = 67.397.601.1999 2001: 2273595031401 = 3.23.29.71.281.56951 2003: 1688214529 = 17.43.1153.2003 2011: 2607162961 = 13.31.2011.3217 2017: 8134561 = 37.109.2017 2021: 301704985 = 5.43.47.73.409 2027: 19794508977127 = 2027.26339.370759 2029: 1773486001 = 7.23.61.89.2029 2031: 268563532783521 = 3.131.677.25541.39521 2033: 4340265931 = 19.43.107.131.379 2039: 3387719303201 = 2039.32609.50951 2045: 301704985 = 5.43.47.73.409 2047: 346808881 = 19.23.37.89.241 2049: 2710945737501 = 3.59.71.131.683.2411 2051: 19884155809 = 7.89.97.293.1123 2053: 54767881 = 7.37.103.2053 2055: 4088711891265 = 3.5.137.233.239.35729 2059: 64774081 = 29.71.163.193 2063: 5686932078001 = 17.31.37.109.1297.2063 2065: 15385904065 = 5.7.37.59.349.577 2069: 440707345 = 5.13.29.113.2069 2071: 188461 = 7.13.19.109 2077: 48628801 = 13.31.67.1801 2081: 17316001 = 53.157.2081 2083: 469593692569 = 2083.8329.27067 2087: 552894301 = 43.61.101.2087 2089: 43620409 = 7.19.157.2089 2091: 3022354401 = 3.11.17.41.101.1301 2093: 27336673 = 7.13.23.37.353 2095: 590754385 = 5.13.109.199.419 2099: 17961771278977 = 29.43.1597.2099.4297 2101: 72108421 = 7.11.191.4903 2103: 126394016001 = 3.11.47.101.701.1151 2111: 891024624721 = 7.11.61.73.1231.2111 2113: 35703361 = 61.277.2113 2117: 10585 = 5.29.73 2119: 1082809 = 7.13.73.163 2123: 37167361 = 7.11.41.61.193 2129: 27160421569 = 17.29.113.229.2129 2131: 776176261 = 7.61.853.2131 2137: 13696033 = 13.17.29.2137 2141: 30189087001 = 73.233.829.2141 2143: 518706721 = 13.43.433.2143 2147: 1569457 = 17.19.43.113 2149: 1033669 = 7.13.37.307 2153: 353670332401 = 137.881.1361.2153 2157: 46748811068961 = 3.41.53.719.881.11321 2159: 34196401 = 17.47.127.337 2161: 88689601 = 7.11.13.41.2161 2165: 161242705 = 5.13.17.337.433 2167: 656601 = 3.11.101.197 2171: 42646887361 = 13.17.19.61.167.997 2173: 17812081 = 7.41.53.1171 2177: 181154701 = 7.13.37.173.311 2179: 593234929 = 7.19.23.89.2179 2183: 977737321 = 11.19.37.59.2143 2185: 1050985 = 5.13.19.23.37 2191: 44238481 = 7.61.313.331 2195: 4382684065 = 5.17.67.439.1753 2201: 382536001 = 31.71.151.1151 2203: 4259100803101 = 23.61.67.131.157.2203 2207: 225593397919 = 2207.6619.15443 2213: 20403012421 = 11.19.31.1423.2213 2215: 60112885 = 5.7.443.3877 2219: 2797002901 = 7.11.19.37.163.317 2221: 1134044821 = 7.13.31.181.2221 2227: 931694401 = 11.17.73.131.521 2229: 227165699598321 = 3.11.29.71.743.1061.4241 2231: 9582145 = 5.23.97.859 2235: 158353658932305 = 3.5.17.89.149.563.83177 2237: 4461725581 = 79.2237.25247 2239: 1223461962241 = 31.149.281.421.2239 2243: 5255104513 = 827.2243.2833 2245: 5632705 = 5.13.193.449 2249: 181154701 = 7.13.37.173.311 2251: 1828377001 = 11.41.1801.2251 2257: 29341 = 13.37.61 2261: 825265 = 5.7.17.19.73 2263: 15841 = 7.31.73 2267: 4351059901 = 31.101.613.2267 2269: 133800661 = 109.541.2269 2273: 152066684449 = 569.2273.117577 2279: 7687547869 = 43.53.149.22639 2281: 93614521 = 7.11.13.41.2281 2283: 975648787041 = 3.107.137.761.29153 2285: 3575798785 = 5.13.17.73.97.457 2287: 5442435649 = 7.73.2287.4657 2291: 18307381 = 29.61.79.131 2293: 26280073 = 73.157.2293 2297: 15906120889 = 1723.2297.4019 2309: 636788093761 = 41.71.211.449.2309 2311: 122785741 = 13.61.67.2311 2315: 3263564305 = 5.53.67.397.463 2317: 44238481 = 7.61.313.331 2319: 765157656692001 = 3.47.53.251.599.773.881 2321: 839275921 = 11.29.37.211.337 2323: 9494101 = 23.61.67.101 2327: 56699735521 = 11.13.61.179.36313 2333: 168659569 = 13.67.83.2333 2335: 3408049527865 = 5.373.467.1399.2797 2339: 11104398708481 = 13.23.31.37.109.127.2339 2341: 115039081 = 157.313.2341 2343: 199438988001 = 3.11.41.71.431.4817 2345: 40280065 = 5.7.67.89.193 2347: 27282539557 = 7.13.139.919.2347 2351: 51436355851 = 2351.4231.5171 2353: 4463641 = 7.13.181.271 2357: 955134181 = 311.1303.2357 2363: 14527315201 = 7.17.31.41.139.691 2369: 6295079737 = 23.73.89.103.409 2371: 44431570261 = 13.397.2371.3631 2377: 146843929 = 163.379.2377 2381: 702683101 = 421.701.2381 2383: 2971847037601 = 13.19.31.271.601.2383 2387: 638959321 = 7.11.13.31.59.349 2389: 62756641 = 109.241.2389 2391: 174470770903594881 = 3.23.41.53.797.12689.115061 [?] 2393: 30194397793 = 7.79.2393.22817 2395: 1348209080065 = 5.19.37.397.479.2017 2397: 97482954489537 = 3.17.47.55109.737969 2399: 12707942017 = 17.463.673.2399 2405: 1050985 = 5.13.19.23.37 2407: 72042143041 = 11.19.29.71.83.2017 2411: 18500666251 = 1447.2411.5303 2413: 12945745 = 5.19.29.37.127 2417: 7281824001 = 3.251.2417.4001 2419: 14483276281 = 19.41.59.181.1741 2423: 52364681461 = 61.347.1021.2423 2427: 1214896993439361 = 3.23.113.197.809.977681 2429: 7492890407905 = 5.7.37.97.347.397.433 2431: 75361 = 11.13.17.31 2435: 33203881585 = 5.7.17.19.37.163.487 2437: 6512378041 = 13.19.31.349.2437 2441: 238244041 = 7.73.191.2441 2443: 638959321 = 7.11.13.31.59.349 2447: 5319173255761 = 113.2447.3271.5881 2449: 867800701 = 31.37.61.79.157 2453: 4077957961 = 7.11.31.47.163.223 2459: 534919693681 = 2459.9833.22123 2461: 4521794641 = 7.13.23.61.107.331 2463: 12996454877121 = 3.17.23.53.89.821.2861 2465: 2465 = 5.17.29 2467: 8346731851 = 823.2467.4111 2471: 27336673 = 7.13.23.37.353 2473: 9261585313 = 337.2473.11113 2477: 9035899110701 = 2477.17333.210461 2479: 403043257 = 19.37.43.67.199 2481: 330794193633 = 3.827.6569.20297 2483: 1825568641 = 13.31.37.191.641 2489: 4340265931 = 19.43.107.131.379 2491: 1070659201 = 17.47.53.131.193 2495: 534570793345 = 5.17.43.109.499.2689 2497: 18382122451 = 7.11.19.227.55351 2501: 252601 = 41.61.101 2503: 235081952731 = 2503.7507.12511 2505: 222215555846145 = 3.5.29.167.449.653.10433 2507: 1162202581 = 23.43.109.10781 2509: 3224065 = 5.13.193.257 2513: 325546585 = 5.7.13.359.1993 2515: 175997185 = 5.7.13.503.769 2517: 12832546007164521 = 3.11.29.71.281.839.801077 2519: 544101481 = 7.11.59.229.523 2521: 101649241 = 61.661.2521 2531: 15297796801 = 7.461.1873.2531 2533: 274569601 = 17.61.149.1777 2537: 17535604465 = 5.17.43.59.233.349 2539: 16851015469 = 19.163.2143.2539 2543: 550067193001 = 487.541.821.2543 2545: 11119105 = 5.17.257.509 2549: 46652524465 = 5.17.197.1093.2549 2551: 683032801 = 11.101.241.2551 2555: 825265 = 5.7.17.19.73 2557: 1954174465 = 5.353.433.2557 2561: 540066241 = 11.13.19.197.1009 2563: 2339165521 = 7.11.233.241.541 2567: 629692801 = 17.31.41.151.193 2569: 295643089 = 7.157.367.733 2571: 435262925087145321 = 3.23.857.2141.35597.96581 [?] 2573: 18785465281 = 11.19.31.83.181.193 2579: 1079942171239 = 2579.18047.23203 2581: 3576804001 = 29.89.101.13721 2587: 670033 = 7.13.37.199 2589: 291226428348047343201 = 3.101.239.401.863.9011.1289621 [?] 2591: 34190968141 = 11.19.103.613.2591 2593: 120981601 = 13.37.97.2593 2595: 947087538769733505 = 3.5.17.23.173.929.2237.449153 [?] 2599: 934784929 = 13.23.73.113.379 2603: 130032865 = 5.19.97.103.137 2609: 111154590001 = 7.13.19.41.601.2609 2611: 46483633 = 7.19.373.937 2615: 1776450565 = 5.317.523.2143 2617: 392807075761 = 19.37.89.2399.2617 2621: 542497201 = 263.787.2621 2623: 10877581 = 11.13.29.43.61 2627: 67994641 = 11.13.37.71.181 2629: 962500561 = 11.239.269.1361 2633: 5599487881 = 13.29.2633.5641 2641: 81926461 = 19.67.139.463 2643: 46748811068961 = 3.41.53.719.881.11321 2647: 8355729313 = 17.19.29.337.2647 2651: 9890881 = 7.11.13.41.241 2657: 3521441665 = 5.7.19.1993.2657 2659: 185602824001 = 7.31.67.2659.4801 2661: 8022231726890001 = 3.101.251.887.1091.109001 2663: 374811729601 = 727.2663.193601 2669: 977892241 = 17.31.53.157.223 2671: 1362132541 = 7.11.37.179.2671 2677: 12801448801 = 29.269.613.2677 2681: 1996385311921 = 7.11.13.17.31.41.241.383 2683: 957044881 = 13.23.1193.2683 2685: 114593508055911048606465 = 3.5.23.179.1409.8297.33287.4768493 [?] 2687: 1123391597833 = 307.953.1429.2687 2689: 238527745 = 5.113.157.2689 2693: 746706961 = 7.11.13.277.2693 2699: 28173681537 = 3.353.2699.9857 2701: 294409 = 37.73.109 2703: 12996454877121 = 3.17.23.53.89.821.2861 2707: 710541481 = 13.61.331.2707 2711: 139592101 = 11.31.151.2711 2713: 3480174001 = 227.2713.5651 2717: 129255841 = 11.13.19.113.421 2719: 15157389061 = 19.181.1621.2719 2723: 82268656561 = 7.13.31.61.389.1229 2729: 33300199505 = 5.17.89.1613.2729 2731: 14913991 = 43.127.2731 2733: 61811815521 = 3.71.113.911.2819 2735: 3637405045 = 5.43.157.197.547 2737: 1597009393 = 7.17.23.73.7993 2739: 3289207215669441 = 3.11.53.83.131.191.821.1103 2741: 12760070401 = 13.113.2741.3169 2743: 3146221 = 13.31.37.211 2747: 975303121 = 13.31.41.67.881 2749: 22665505 = 5.17.97.2749 2753: 75765313 = 13.29.73.2753 2755: 449065 = 5.19.29.163 2759: 3119101921 = 13.19.23.31.89.199 2761: 188689501 = 7.11.13.251.751 2765: 5368969165 = 5.7.53.79.36637 2767: 5487578041 = 11.29.2767.6217 2771: 545363281 = 17.23.43.163.199 2773: 7092343919761 = 47.59.3037.842161 2777: 4216799521 = 11.31.61.73.2777 2779: 13992265 = 5.7.19.53.397 2785: 20910960976465 = 5.7.157.223.557.30637 2787: 3713121557601 = 3.17.131.281.929.2129 2789: 13654188181 = 37.307.431.2789 2791: 194675041 = 11.17.373.2791 2795: 115542505 = 5.13.43.67.617 2797: 49237316749 = 1399.2797.12583 2801: 1348964401 = 401.1201.2801 2803: 642716875801 = 7.11.13.37.41.151.2803 2807: 241242001 = 7.11.13.401.601 2813: 612347905 = 5.13.17.29.97.197 2815: 2652506486785 = 5.463.563.1009.2017 2819: 61811815521 = 3.71.113.911.2819 2821: 2821 = 7.13.31 2823: 28148013211641 = 3.29.41.941.2423.3461 2827: 1031750401 = 11.31.61.193.257 2829: 1772996733201 = 3.23.41.89.101.113.617 2831: 12261061 = 19.61.71.149 2833: 545570641 = 7.11.41.61.2833 2837: 479252075149 = 67.127.2837.19853 2839: 42646887361 = 13.17.19.61.167.997 2841: 607368553651905 = 3.5.107.947.15137.26399 2843: 261890754721 = 17.61.211.421.2843 2845: 251841872305 = 5.67.569.1033.1279 2849: 672389641 = 7.11.37.53.61.73 2851: 3030758401 = 11.241.401.2851 2857: 1672719217 = 13.29.1553.2857 2859: 7326285446481 = 3.23.53.281.953.7481 2861: 9139810681 = 859.2861.3719 2863: 602074585 = 5.7.137.307.409 2867: 4852062061 = 7.11.31.47.61.709 2869: 78120001 = 19.73.151.373 2879: 8956911601 = 11.17.127.131.2879 2881: 115542505 = 5.13.43.67.617 2885: 84417985 = 5.29.577.1009 2887: 9739972945 = 5.17.19.2089.2887 2893: 29483017201 = 11.31.263.547.601 2897: 128295478801 = 7.37.163.1049.2897 2899: 2199700321 = 7.13.61.223.1777 2903: 733547013841 = 2903.8707.29021 2905: 18390744505 = 5.7.83.1453.4357 2909: 41983866145 = 5.13.17.37.353.2909 2911: 548871961 = 11.41.61.71.281 2913: 5587772812161 = 3.23.29.353.971.8147 2917: 12299638429 = 13.19.43.397.2917 2921: 2342644921 = 23.31.41.127.631 2923: 8719309 = 19.37.79.157 2927: 20871450601 = 463.2927.15401 2929: 58489201 = 19.29.101.1051 2931: 1299934624255521 = 3.59.881.977.2411.3539 2933: 1615335085 = 5.7.13.37.229.419 2935: 510675333805 = 5.7.13.587.1087.1759 2939: 2583708107761 = 11.31.41.227.277.2939 2941: 4752717761 = 11.17.107.173.1373 2949: 13053581557039793157 = 3.173.983.9203.27179.102293 [?] 2951: 27360228961 = 13.17.31.73.227.241 2953: 897880321 = 13.19.1231.2953 2955: 77403893505 = 3.5.197.227.257.449 2957: 2159003281 = 17.29.1481.2957 2959: 962500561 = 11.239.269.1361 2963: 33604861537 = 13.17.19.37.73.2963 2965: 28787185 = 5.7.19.73.593 2969: 484662529 = 13.29.433.2969 2971: 26474581 = 7.19.67.2971 2977: 18900973 = 7.13.229.907 2981: 658801 = 11.13.17.271 2983: 8719309 = 19.37.79.157 2987: 17163788881 = 29.103.113.211.241 2993: 410041 = 41.73.137 2995: 15403285 = 5.37.139.599 2999: 63237358424809 = 2999.137909.152899This list has been extended (in December 2007) to all divisors below 10000.
We also have (since 2003) a complete list of the least Carmichael multiples of all odd primes below 10000.
I put online the first version of the above table in 2003 (Gérard P. Michon, 2003-12-01). At first, it was strictly a compilation of the smallest Carmichael multiples (with 16 digits or less) of all divisors up to 2999.
An integer can have a Carmichael multiple only if it's an odd number coprime to its Euler totient (i.e., a cyclic number other than 2). Many years ago, I formulated the conjecture that this necessary condition was also sufficient (i.e., all odd cyclic numbers have Carmichael multiples ) and the table was meant to provide computational support for that conjecture. Among all such numbers below 3000, the original table featured only 8 gaps (885, 2391, 2517, 2571, 2589, 2595, 2685 and 2949). This is how it remained for over 4 years...
On 2007-08-04, Max Alekseyev ("Maxal") kindly credited me with that conjecture in the Open Problem Garden and challenged people to plug those gaps. As chronicled below, that challenge was met in December 2007 by Joe K. Crump (starting with the divisor 2517). Joe used tools which need not produce the smallest Carmichael multiple of a given divisor.
In the above table, a question mark [?] is used to indicate that the Carmichael number listed need not be the smallest Carmichael multiple of the relevant divisor. No such mark is needed for the Carmichael multiple of 2517 first discovered by Joe Crump, though... Once Joe had paved the way, it took us only a few hours (on 2007-12-22) to establish that it's indeed the smallest one there is (it happens to be only 28% above the arbitrary cutoff limit we used in 2003).
| Divisor | Carmichael Multiple(s) | Author | Date | |
|---|---|---|---|---|
| 885: | 24354644805191195265 | Joe Crump | 2007-12-21 | |
| 2391: | 174470770903594881 | 582506871340419201 | Joe Crump | 2007-12-21 |
| 2517: | 12832546007164521 | Joe Crump | 2007-12-20 | |
| 2571: | 435262925087145321 | 1340771505059480001 | Joe Crump | 2007-12-21 |
| 2589: | 291226428348047343201 | 401998541941991167041 | Joe Crump | 2007-12-21 |
| 2595: | 947087538769733505 | Joe Crump | 2007-12-21 | |
| 2685: | 114593508055911048606465 | Joe Crump | 2007-12-24 | |
| 2949: | 13053581557039793157 | Joe Crump | 2007-12-21 | |
