414:
1377:
I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the
441:
in hospital. In their conversation, Hardy stated that the number 1729 from a taxicab he rode was a "dull" number and "hopefully it is not unfavourable omen", but
Ramanujan otherwise stated it is a number that can be expressed as the
378:. Investigating pairs of its distinct integer-valued that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible
909:
682:
1554:
322:
1592:
528:
488:
1175:
1014:
979:
579:
944:
393:(a number that counts the points in a three-dimensional pattern formed by a point surrounded by concentric cubical layers of points), the nineteenth
710:
1599:
1290:
1271:
1252:
1021:
800:
292:, meaning its factors are 1, 7, 13, 19, 91, 133, 247, and 1729. It is the multiplication of its first three smallest prime numbers
1476:
1415:
1318:
1226:
1055:
766:
364:
1638:
533:
1729 was also found in one of
Ramanujan's notebooks dated years before the incident and was noted by French mathematician
1664:
1214:
718:
537:
in 1657. A commemorative plaque now appears at the site of the
RamanujanâHardy incident, at 2 Colinette Road in
328:, and specifically the first ChernickâCarmichael number. Furthermore, it is the first in the family of absolute
1432:
544:
The same expression defines 1729 as the first in the sequence of "Fermat near misses" defined, in reference to
1344:
1685:
843:
616:
1500:
295:
1368:
1559:
335:
1729 can be defined by summing each of its digits, multiplying by the resulting number with its digit
446:
in two different ways. This conversation in the aftermath led to a new class of numbers known as the
545:
493:
453:
1142:
179:
1458:
1395:
1338:
1308:
1092:
1072:
1037:
756:
739:
1372:
984:
949:
783:
551:
914:
417:
1729 can be expressed as a sum of two positive cubes in two ways, illustrated geometrically.
1622:
390:
534:
8:
1138:
438:
277:
394:
368:
1400:
Truth, Existence and
Explanation: FilMat 2016 Studies in the Philosophy of Mathematics
1113:
1680:
1650:
1619:
1472:
1411:
1314:
1222:
1210:
1051:
762:
714:
360:
329:
325:
1362:
820:
1464:
1403:
1379:
1202:
1098:
1043:
815:
724:
443:
402:
336:
289:
261:
104:
386:
114:
63:
50:
1194:
447:
397:(a figurate number in which the arrangement of points resembles the shape of a
375:
340:
257:
249:
156:
146:
1468:
1407:
1206:
1047:
1674:
356:
344:
253:
166:
126:
93:
90:
87:
84:
81:
78:
75:
72:
69:
29:
379:
1646:
1492:
1358:
1280:
1261:
1242:
835:
434:
352:
273:
231:
218:
1627:
1238:
Other sources on its figurate numbers can be found in the following:
398:
1201:. Problem Books in Mathematics, Volume 1 (3rd ed.). Springer.
430:
1340:
Number Theory and
Geometry: An Introduction to Arithmetic Geometry
613:
expressed
Carmichael number as the product of three prime numbers
136:
55:
1266:"Sequence A051624 (12-gonal (or dodecagonal) numbers)"
413:
1617:
1497:"Sequence A050794 (Consider the Diophantine equation
538:
192:
96:
1665:
Why does the number 1729 show up in so many
Futurama episodes?
707:
Table of divisors of all the natural numbers from 1. to 10000
348:
205:
581:, which are also expressible as the sum of two other cubes.
1496:
1284:
1265:
1246:
840:"Sequence A033502 (Carmichael number of the form
839:
590:
1114:"We've found a quicker way to multiply really big numbers"
1491:
1279:
1260:
1241:
834:
1562:
1503:
1145:
987:
952:
917:
846:
619:
554:
496:
456:
298:
66:
1247:"Sequence A005898 (Centered cube numbers)"
741:
Elementary Theory of
Numbers: Second English Edition
1137:Harvey, David; Hoeven, Joris van der (March 2019).
1586:
1548:
1169:
1008:
973:
938:
903:
676:
573:
522:
482:
450:. 1729 is the second taxicab number, expressed as
316:
1639:"1729: Taxi Cab Number or Hardy-Ramanujan Number"
1672:
1460:My Search for Ramanujan: How I Learned to Count
1433:"A black plaque for Ramanujan, Hardy and 1,729"
1285:"Sequence A051876 (24-gonal numbers)"
264:in two different ways. It is also known as the
256:and preceding 1730. It is the first nontrivial
1636:
1398:. In Piazza, Mario; Pulcini, Gabriele (eds.).
1336:
828:
367:two numbers is based. This is an example of a
1232:
761:(2nd ed.). Academic Press. p. 340.
1136:
1086:
1084:
1485:
738:Sierpinski, W. (1998). Schinzel, A. (ed.).
1306:
758:Elementary Number Theory with Applications
737:
1600:On-Line Encyclopedia of Integer Sequences
1332:
1330:
1291:On-Line Encyclopedia of Integer Sequences
1272:On-Line Encyclopedia of Integer Sequences
1253:On-Line Encyclopedia of Integer Sequences
1081:
1039:A Concrete Introduction to Higher Algebra
1022:On-Line Encyclopedia of Integer Sequences
819:
731:
363:on which the fastest known algorithm for
121:(one thousand seven hundred twenty-ninth)
1456:
1430:
1091:Deza, Michel-marie; Deza, Elena (2012).
1090:
1064:
798:
775:
610:
596:
412:
408:
1042:(2nd ed.). Springer. p. 409.
792:
1673:
1431:Marshall, Michael (24 February 2017).
1327:
1105:
1035:
808:Bulletin American Mathematical Society
704:
343:. This property can be found in other
283:
109:one thousand seven hundred twenty-nine
1618:
1393:
1387:
1357:
1351:
1307:Edward, Graham; Ward, Thomas (2005).
1029:
754:
698:
437:when he visited Indian mathematician
385:Visually, 1729 can be found in other
1070:
904:{\displaystyle (6k+1)(12k+1)(18k+1)}
781:
748:
677:{\displaystyle (6k+1)(12k+1)(18k+1)}
1549:{\displaystyle x^{3}+y^{3}=z^{3}+1}
1193:
1187:
785:Mersenne Numbers And Fermat Numbers
317:{\displaystyle 7\times 13\times 19}
13:
1450:
1424:
1300:
1199:Unsolved Problems in Number Theory
1130:
1111:
332:, a subset of Carmichael numbers.
14:
1697:
1611:
1457:Ono, Ken; Aczel, Amir D. (2016).
405:and the seventh 84-gonal number.
382:of a four-variable pair is 1729.
355:. However, this does not work on
1587:{\displaystyle 1<x<y<z}
1310:An Introduction to Number Theory
1139:"Integer multiplication in time
1077:. World Scientific. p. 411.
141:1, 7, 13, 19, 91, 133, 247, 1729
1337:Lozano-Robledo, Ălvaro (2019).
821:10.1090/S0002-9904-1939-06953-X
788:. World Scientific. p. 51.
1164:
1149:
1003:
988:
968:
953:
933:
918:
898:
883:
880:
865:
862:
847:
671:
656:
653:
638:
635:
620:
603:
1:
1637:Grime, James; Bowley, Roger.
1345:American Mathematical Society
744:. North-Holland. p. 233.
691:
433:of the British mathematician
374:1729 can be expressed as the
359:. It is the dimension of the
324:. Relatedly, it is the third
1074:Perfect And Amicable Numbers
801:"On Fermat's simple theorem"
523:{\displaystyle 9^{3}+10^{3}}
483:{\displaystyle 1^{3}+12^{3}}
427:Hardy–Ramanujan number
7:
1594:) or 'Fermat near misses')"
1036:Childs, Lindsay N. (1995).
584:
10:
1702:
1493:Sloane, N. J. A.
1396:"Structure and Structures"
1369:Cambridge University Press
1281:Sloane, N. J. A.
1262:Sloane, N. J. A.
1243:Sloane, N. J. A.
1170:{\displaystyle O(n\log n)}
836:Sloane, N. J. A.
1469:10.1007/978-3-319-25568-2
1408:10.1007/978-3-319-93342-9
1313:. Springer. p. 117.
1207:10.1007/978-0-387-26677-0
1048:10.1007/978-1-4419-8702-0
548:, as numbers of the form
230:
217:
204:
191:
178:
165:
155:
145:
135:
125:
113:
103:
45:
24:
1623:"HardyâRamanujan Number"
1394:Kahle, Reinhard (2018).
609:It is a number in which
444:sum of two cubic numbers
262:sum of two cubic numbers
1382:in two different ways."
1009:{\displaystyle (18k+1)}
974:{\displaystyle (12k+1)}
574:{\displaystyle 1+z^{3}}
1588:
1550:
1171:
1010:
975:
940:
939:{\displaystyle (6k+1)}
905:
755:Koshy, Thomas (2007).
705:Anjema, Henry (1767).
678:
575:
524:
484:
421:1729 is also known as
418:
318:
270:HardyâRamanujan number
16:Hardy-Ramanujan number
1589:
1551:
1172:
1011:
976:
941:
906:
799:Chernick, J. (1939).
679:
597:Explanatory footnotes
576:
546:Fermat's Last Theorem
525:
485:
416:
409:As a Ramanujan number
401:), the thirteenth 24-
319:
1560:
1501:
1143:
1071:Deza, Elena (2023).
985:
950:
915:
844:
782:Deza, Elena (2022).
617:
552:
494:
454:
391:centered cube number
296:
1686:Srinivasa Ramanujan
1016:are prime numbers)"
439:Srinivasa Ramanujan
337:permutably switched
284:As a natural number
278:Srinivasa Ramanujan
260:, expressed as the
1620:Weisstein, Eric W.
1603:. OEIS Foundation.
1584:
1546:
1294:. OEIS Foundation.
1275:. OEIS Foundation.
1256:. OEIS Foundation.
1167:
1025:. OEIS Foundation.
1006:
971:
936:
901:
674:
571:
520:
480:
419:
395:dodecagonal number
389:. It is the tenth
369:galactic algorithm
330:Euler pseudoprimes
314:
1478:978-3-319-25568-2
1417:978-3-319-93342-9
1320:978-1-85233-917-3
1227:978-0-387-26677-0
1057:978-1-4419-8702-0
768:978-0-12-372487-8
535:FrĂŠnicle de Bessy
429:, named after an
361:Fourier transform
326:Carmichael number
243:
242:
41:
40:
1693:
1661:
1659:
1658:
1649:. Archived from
1633:
1632:
1605:
1604:
1593:
1591:
1590:
1585:
1555:
1553:
1552:
1547:
1539:
1538:
1526:
1525:
1513:
1512:
1489:
1483:
1482:
1454:
1448:
1447:
1445:
1443:
1428:
1422:
1421:
1391:
1385:
1384:
1380:sum of two cubes
1355:
1349:
1348:
1334:
1325:
1324:
1304:
1298:
1295:
1276:
1257:
1236:
1230:
1220:
1191:
1185:
1184:
1176:
1174:
1173:
1168:
1134:
1128:
1127:
1125:
1124:
1109:
1103:
1102:
1099:World Scientific
1094:Figurate Numbers
1088:
1079:
1078:
1068:
1062:
1061:
1033:
1027:
1026:
1015:
1013:
1012:
1007:
980:
978:
977:
972:
945:
943:
942:
937:
910:
908:
907:
902:
832:
826:
825:
823:
805:
796:
790:
789:
779:
773:
772:
752:
746:
745:
735:
729:
728:
725:Internet Archive
723:– via the
702:
685:
683:
681:
680:
675:
607:
580:
578:
577:
572:
570:
569:
529:
527:
526:
521:
519:
518:
506:
505:
489:
487:
486:
481:
479:
478:
466:
465:
423:Ramanujan number
387:figurate numbers
323:
321:
320:
315:
266:Ramanujan number
26:
25:
22:
21:
1701:
1700:
1696:
1695:
1694:
1692:
1691:
1690:
1671:
1670:
1656:
1654:
1614:
1609:
1608:
1561:
1558:
1557:
1534:
1530:
1521:
1517:
1508:
1504:
1502:
1499:
1498:
1490:
1486:
1479:
1463:. p. 228.
1455:
1451:
1441:
1439:
1429:
1425:
1418:
1402:. p. 115.
1392:
1388:
1356:
1352:
1335:
1328:
1321:
1305:
1301:
1237:
1233:
1221:
1217:
1195:Guy, Richard K.
1192:
1188:
1183:. hal-02070778.
1144:
1141:
1140:
1135:
1131:
1122:
1120:
1112:Harvey, David.
1110:
1106:
1089:
1082:
1069:
1065:
1058:
1034:
1030:
986:
983:
982:
951:
948:
947:
916:
913:
912:
845:
842:
841:
833:
829:
803:
797:
793:
780:
776:
769:
753:
749:
736:
732:
721:
703:
699:
694:
689:
688:
618:
615:
614:
611:Chernick (1939)
608:
604:
599:
587:
565:
561:
553:
550:
549:
514:
510:
501:
497:
495:
492:
491:
474:
470:
461:
457:
455:
452:
451:
411:
297:
294:
293:
286:
239:
226:
213:
200:
187:
174:
120:
99:
61:
60:
51:List of numbers
20:
17:
12:
11:
5:
1699:
1689:
1688:
1683:
1669:
1668:
1662:
1634:
1613:
1612:External links
1610:
1607:
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1507:
1484:
1477:
1449:
1423:
1416:
1386:
1350:
1347:. p. 413.
1326:
1319:
1299:
1297:
1296:
1277:
1258:
1231:
1215:
1186:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1129:
1104:
1101:. p. 436.
1080:
1063:
1056:
1028:
1005:
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996:
993:
990:
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967:
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961:
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882:
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864:
861:
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855:
852:
849:
827:
814:(4): 269â274.
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747:
730:
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568:
564:
560:
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517:
513:
509:
504:
500:
477:
473:
469:
464:
460:
448:taxicab number
410:
407:
376:quadratic form
347:, such as the
345:number systems
341:harshad number
313:
310:
307:
304:
301:
285:
282:
272:, named after
258:taxicab number
250:natural number
241:
240:
237:
234:
228:
227:
224:
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211:
208:
202:
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107:
101:
100:
62:
59:
58:
53:
47:
46:
43:
42:
39:
38:
35:
32:
19:Natural number
18:
15:
9:
6:
4:
3:
2:
1698:
1687:
1684:
1682:
1679:
1678:
1676:
1666:
1663:
1653:on 2017-03-06
1652:
1648:
1644:
1640:
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1616:
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1602:
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1505:
1494:
1488:
1480:
1474:
1470:
1466:
1462:
1461:
1453:
1438:
1437:Good Thinking
1434:
1427:
1419:
1413:
1409:
1405:
1401:
1397:
1390:
1383:
1381:
1374:
1370:
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1278:
1274:
1273:
1267:
1263:
1259:
1255:
1254:
1248:
1244:
1240:
1239:
1235:
1228:
1224:
1218:
1216:0-387-20860-7
1212:
1208:
1204:
1200:
1196:
1190:
1182:
1178:
1161:
1158:
1155:
1152:
1146:
1133:
1119:
1115:
1108:
1100:
1096:
1095:
1087:
1085:
1076:
1075:
1067:
1059:
1053:
1049:
1045:
1041:
1040:
1032:
1024:
1023:
1017:
1000:
997:
994:
991:
965:
962:
959:
956:
930:
927:
924:
921:
895:
892:
889:
886:
877:
874:
871:
868:
859:
856:
853:
850:
837:
831:
822:
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802:
795:
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759:
751:
743:
742:
734:
726:
722:
720:9781140919421
716:
712:
708:
701:
697:
668:
665:
662:
659:
650:
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632:
629:
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623:
612:
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582:
566:
562:
558:
555:
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542:
540:
536:
531:
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507:
502:
498:
475:
471:
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458:
449:
445:
440:
436:
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428:
424:
415:
406:
404:
400:
396:
392:
388:
383:
381:
377:
372:
370:
366:
362:
358:
357:binary number
354:
350:
346:
342:
338:
333:
331:
327:
311:
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209:
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196:
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183:
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177:
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160:
158:
157:Roman numeral
154:
150:
148:
147:Greek numeral
144:
140:
138:
134:
130:
128:
127:Factorization
124:
118:
116:
112:
108:
106:
102:
98:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
65:
57:
54:
52:
49:
48:
44:
36:
33:
31:
30:← 1728
28:
27:
23:
1655:. Retrieved
1651:the original
1642:
1626:
1597:
1487:
1459:
1452:
1440:. Retrieved
1436:
1426:
1399:
1389:
1376:
1367:. New York:
1363:
1359:Hardy, G. H.
1353:
1339:
1309:
1302:
1288:
1269:
1250:
1234:
1198:
1189:
1180:
1132:
1121:. Retrieved
1117:
1107:
1093:
1073:
1066:
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380:discriminant
373:
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1647:Brady Haran
1643:Numberphile
435:G. H. Hardy
365:multiplying
353:hexadecimal
274:G. H. Hardy
232:Hexadecimal
171:11011000001
131:7 Ă 13 Ă 19
1675:Categories
1657:2013-04-02
1371:. p.
1123:2021-11-01
709:. p.
692:References
252:following
219:Duodecimal
1667:, io9.com
1628:MathWorld
1364:Ramanujan
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309:×
303:×
290:composite
1681:Integers
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1197:(2004).
1118:phys.org
911:, where
585:See also
431:anecdote
288:1729 is
161:MDCCXXIX
137:Divisors
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180:Ternary
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490:and
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246:1729
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