70:). For instance, HP(10) = 773, as 10 factors as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation
191:
through 100 other than the ones still unresolved. It also included a now-obsolete list of 3-digit unresolved numbers (The 58 listed have been cut precisely in half as of August 2012). It appears that this article is largely responsible for provoking attempts by others to resolve the case involving
89:
The outstanding computational problem as of 2016 is whether HP(49) = HP(77) can be calculated in practice. As each iteration is greater than the previous up until a prime is reached, factorizations generally grow more difficult so long as an end is not reached. As of August 2016 the
94:
factor of HP49(119) after a break was achieved on 3 December 2014 with the calculation of HP49(117). This followed the factorization of HP49(110) on 8 September 2012 and of HP49(104) on 11 January 2011, and prior calculations extending for the larger part of a decade that made extensive use of
212:. The brief article does little other than state the origins of the subject, define terms, give a couple of examples, mention machinery and methods used at the time, and then provide tables. It appears that Mr. De Geest is responsible for the notation now in use. The
307:"Sequence A037274 (Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached))"
148:
terms, the existence has probability 1 for all numbers, but such heuristics make assumptions about numbers drawn from a wide variety of processes that, though they are likely correct, fall short of the standard of
171:
While it is unlikely that the idea was not conceived of numerous times in the past, the first reference in print appears to be an article written in 1990 in a small and now-defunct publication called
74:
for the homeprime, leaving out parentheses. Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient
126:
2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, ... (sequence
1143:
95:
computational resources. Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at
375:
746:
1649:
312:
229:
133:
828:
751:
665:
108:
176:
368:
1002:
1083:
361:
1205:
863:
1230:
696:
244:
47:
1644:
1138:
220:
as the term for the number of numbers, including the prime itself, that have a certain prime as its home prime.
175:. The same person who authored that article, Jeffrey Heleen, revisited the subject in the 1996–7 volume of the
140:
Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of
83:
771:
239:
205:
200:
to describe composites and the primes that they lead to, with numbers leading to the same home prime called
204:(even if one is an iterate of another), and calls the number of iterations required to reach a parent, the
1288:
417:
1625:
1215:
868:
776:
43:
96:
1195:
100:
1190:
848:
234:
1298:
1235:
1225:
1210:
843:
701:
150:
622:
1267:
1242:
1220:
1200:
823:
795:
488:
282:
264:
1177:
1167:
1162:
1099:
946:
813:
716:
91:
79:
144:
of a home prime for any specific number might entail its effective computation. In purely
878:
838:
721:
686:
650:
605:
458:
446:
1283:
1257:
1154:
1022:
873:
833:
818:
690:
581:
546:
501:
426:
408:
1638:
1293:
1058:
922:
895:
731:
596:
534:
525:
510:
473:
399:
17:
1614:
1609:
1604:
1599:
1594:
1589:
1584:
1579:
1574:
1569:
1564:
1559:
1554:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1514:
1509:
1504:
1499:
1494:
1489:
1484:
1479:
1474:
1469:
1464:
1459:
1454:
1449:
1444:
1247:
970:
853:
736:
726:
711:
706:
670:
384:
51:
39:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1389:
1384:
1379:
1374:
1369:
1364:
1359:
1354:
1349:
1344:
1339:
1185:
858:
766:
761:
741:
655:
558:
434:
330:
302:
1262:
1078:
986:
906:
756:
660:
325:
1303:
1252:
1133:
145:
141:
75:
320:
335:
J. Heleen, Family
Numbers: Constructing Primes By Prime Factor Splicing,
805:
353:
112:
32:
800:
786:
213:
104:
357:
181:
Family
Numbers: Constructing Primes By Prime Factor Splicing
90:
pursuit of HP(49) concerns the factorization of a 251-digit
306:
128:
1334:
1329:
1324:
1319:
58:
th intermediate stage in the process of determining HP(
346:
J. Heleen, Family
Numbers: Mathemagical Black Holes,
208:
under the map to obtain a home prime, the number of
1312:
1276:
1176:
1153:
1127:
894:
887:
785:
679:
643:
392:
115:and also has lists for the bases 2 through 9.
111:maintains the complete known data through 1000 in
1016: = 0, 1, 2, 3, ...
369:
8:
192:49 and 77. The article uses the terms
326:http://mathworld.wolfram.com/HomePrime.html
891:
376:
362:
354:
348:Recreational and Educational Computing, 5
313:On-Line Encyclopedia of Integer Sequences
230:List of recreational number theory topics
153:usually required of mathematical claims.
321:http://www.worldofnumbers.com/topic1.htm
173:Recreational and Educational Computation
167:Early history and additional terminology
255:
183:, which included all of the results HP(
7:
109:Great Internet Mersenne Prime Search
177:Journal of Recreational Mathematics
82:, but the subject is really one in
14:
1650:Base-dependent integer sequences
752:Supersingular (moonshine theory)
331:Home Prime Search on Prime Wiki
747:Supersingular (elliptic curve)
107:primarily associated with the
1:
528:2 ± 2 ± 1
281:WraithX (8 September 2012).
54:including repetitions. The
263:WraithX (3 December 2014).
245:Concatenation (mathematics)
1666:
303:Sloane, N. J. A.
1623:
1134:Mega (1,000,000+ digits)
1003:Arithmetic progression (
343:, pp. 116–9, 1996-7
84:recreational mathematics
240:Persistence of a number
206:persistence of a number
179:in an article entitled
42:obtained by repeatedly
1289:Industrial-grade prime
666:Newman–Shanks–Williams
162:HP(n) = n for n prime.
38:greater than 1 is the
1626:List of prime numbers
1084:Sophie Germain/Safe (
808:(10 − 1)/9
1117: ± 7, ...
644:By integer sequence
429:(2 + 1)/3
350::5, p. 6, 1990
235:Prime factorization
1299:Formula for primes
932: + 2 or
864:Smarandache–Wellin
316:. OEIS Foundation.
1645:Integer sequences
1632:
1631:
1243:Carmichael number
1178:Composite numbers
1113: ± 3, 8
1109: ± 1, 4
1072: ± 1, …
1068: ± 1, 4
1064: ± 1, 2
1054:
1053:
599:3·2 − 1
504:2·3 + 1
418:Double Mersenne (
286:mersenneforum.org
268:mersenneforum.org
118:The primes in HP(
80:composite numbers
1657:
1163:Eisenstein prime
1118:
1094:
1073:
1045:
1017:
997:
981:
965:
960: + 6,
956: + 2,
941:
936: + 4,
917:
892:
809:
772:Highly cototient
634:
633:
627:
617:
600:
591:
576:
553:
552:·2 − 1
541:
540:·2 + 1
529:
520:
505:
496:
483:
468:
453:
441:
440:·2 + 1
430:
421:
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403:
378:
371:
364:
355:
317:
290:
289:
278:
272:
271:
260:
131:
97:Patrick De Geest
62:) is designated
1665:
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1660:
1659:
1658:
1656:
1655:
1654:
1635:
1634:
1633:
1628:
1619:
1313:First 60 primes
1308:
1272:
1172:
1155:Complex numbers
1149:
1123:
1101:
1085:
1060:
1059:Bi-twin chain (
1050:
1024:
1004:
988:
972:
948:
924:
908:
883:
869:Strobogrammatic
807:
781:
675:
639:
631:
625:
624:
607:
598:
583:
560:
548:
536:
527:
512:
503:
490:
482:# + 1
480:
475:
467:# ± 1
465:
460:
452:! ± 1
448:
436:
428:
420:2 − 1
419:
411:2 − 1
410:
402:2 + 1
401:
388:
382:
301:
298:
293:
280:
279:
275:
262:
261:
257:
253:
226:
169:
159:
127:
46:the increasing
12:
11:
5:
1663:
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1316:
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1301:
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1284:Probable prime
1280:
1278:
1277:Related topics
1274:
1273:
1271:
1270:
1265:
1260:
1258:Sphenic number
1255:
1250:
1245:
1240:
1239:
1238:
1233:
1228:
1223:
1218:
1213:
1208:
1203:
1198:
1193:
1182:
1180:
1174:
1173:
1171:
1170:
1168:Gaussian prime
1165:
1159:
1157:
1151:
1150:
1148:
1147:
1146:
1136:
1131:
1129:
1125:
1124:
1122:
1121:
1097:
1093: + 1
1081:
1076:
1055:
1052:
1051:
1049:
1048:
1020:
1000:
996: + 6
984:
980: + 4
968:
964: + 8
944:
940: + 6
920:
916: + 2
903:
901:
889:
885:
884:
882:
881:
876:
871:
866:
861:
856:
851:
846:
841:
836:
831:
826:
821:
816:
811:
803:
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792:
790:
783:
782:
780:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
683:
681:
677:
676:
674:
673:
668:
663:
658:
653:
647:
645:
641:
640:
638:
637:
620:
616: − 1
603:
594:
579:
556:
544:
532:
523:
508:
499:
495: + 1
486:
478:
471:
463:
456:
444:
432:
424:
415:
406:
396:
394:
390:
389:
383:
381:
380:
373:
366:
358:
352:
351:
344:
333:
328:
323:
318:
297:
294:
292:
291:
283:"HP49(100)..."
273:
265:"HP49(100)..."
254:
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242:
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222:
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158:
155:
138:
137:
101:worldofnumbers
78:for factoring
13:
10:
9:
6:
4:
3:
2:
1662:
1651:
1648:
1646:
1643:
1642:
1640:
1627:
1622:
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1613:
1611:
1608:
1606:
1603:
1601:
1598:
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1573:
1571:
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1506:
1503:
1501:
1498:
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1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1451:
1448:
1446:
1443:
1441:
1438:
1436:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1348:
1346:
1343:
1341:
1338:
1336:
1333:
1331:
1328:
1326:
1323:
1321:
1318:
1317:
1315:
1311:
1305:
1302:
1300:
1297:
1295:
1294:Illegal prime
1292:
1290:
1287:
1285:
1282:
1281:
1279:
1275:
1269:
1266:
1264:
1261:
1259:
1256:
1254:
1251:
1249:
1246:
1244:
1241:
1237:
1234:
1232:
1229:
1227:
1224:
1222:
1219:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1188:
1187:
1184:
1183:
1181:
1179:
1175:
1169:
1166:
1164:
1161:
1160:
1158:
1156:
1152:
1145:
1142:
1141:
1140:
1139:Largest known
1137:
1135:
1132:
1130:
1126:
1120:
1116:
1112:
1108:
1104:
1098:
1096:
1092:
1088:
1082:
1080:
1077:
1075:
1071:
1067:
1063:
1057:
1056:
1047:
1044:
1041: +
1040:
1036:
1032:
1029: −
1028:
1021:
1019:
1015:
1011:
1008: +
1007:
1001:
999:
995:
991:
985:
983:
979:
975:
969:
967:
963:
959:
955:
951:
945:
943:
939:
935:
931:
927:
921:
919:
915:
911:
905:
904:
902:
900:
898:
893:
890:
886:
880:
877:
875:
872:
870:
867:
865:
862:
860:
857:
855:
852:
850:
847:
845:
842:
840:
837:
835:
832:
830:
827:
825:
822:
820:
817:
815:
812:
810:
804:
802:
799:
797:
794:
793:
791:
788:
784:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
692:
688:
685:
684:
682:
678:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
648:
646:
642:
636:
630:
621:
619:
615:
611:
604:
602:
595:
593:
590:
587: +
586:
580:
578:
575:
572: −
571:
567:
564: −
563:
557:
555:
551:
545:
543:
539:
533:
531:
524:
522:
519:
516: +
515:
509:
507:
500:
498:
494:
489:Pythagorean (
487:
485:
481:
472:
470:
466:
457:
455:
451:
445:
443:
439:
433:
431:
425:
423:
416:
414:
407:
405:
398:
397:
395:
391:
386:
379:
374:
372:
367:
365:
360:
359:
356:
349:
345:
342:
338:
337:J. Rec. Math.
334:
332:
329:
327:
324:
322:
319:
315:
314:
308:
304:
300:
299:
295:
287:
284:
277:
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135:
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110:
106:
102:
98:
93:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
52:prime factors
49:
48:concatenation
45:
41:
37:
34:
30:
26:
23:
19:
18:number theory
1248:Almost prime
1206:Euler–Jacobi
1114:
1110:
1106:
1102:
1100:Cunningham (
1090:
1086:
1069:
1065:
1061:
1042:
1038:
1034:
1030:
1026:
1025:consecutive
1013:
1009:
1005:
993:
989:
977:
973:
961:
957:
953:
949:
947:Quadruplet (
937:
933:
929:
925:
913:
909:
896:
844:Full reptend
702:Wolstenholme
697:Wall–Sun–Sun
628:
613:
609:
588:
584:
573:
569:
565:
561:
549:
537:
517:
513:
492:
476:
461:
449:
437:
385:Prime number
347:
340:
336:
310:
285:
276:
267:
258:
217:
209:
201:
197:
193:
188:
184:
180:
172:
170:
139:
119:
117:
103:website. A
88:
71:
67:
63:
59:
55:
40:prime number
35:
28:
24:
21:
15:
1231:Somer–Lucas
1186:Pseudoprime
824:Truncatable
796:Palindromic
680:By property
459:Primorial (
447:Factorial (
1639:Categories
1268:Pernicious
1263:Interprime
1023:Balanced (
814:Permutable
789:-dependent
606:Williams (
502:Pierpont (
427:Wagstaff
409:Mersenne (
393:By formula
296:References
218:homeliness
216:also uses
157:Properties
76:algorithms
22:home prime
1304:Prime gap
1253:Semiprime
1216:Frobenius
923:Triplet (
722:Ramanujan
717:Fortunate
687:Wieferich
651:Fibonacci
582:Leyland (
547:Woodall (
526:Solinas (
511:Quartan (
146:heuristic
142:existence
92:composite
44:factoring
1196:Elliptic
971:Cousin (
888:Patterns
879:Tetradic
874:Dihedral
839:Primeval
834:Delicate
819:Circular
806:Repunit
597:Thabit (
535:Cullen (
474:Euclid (
400:Fermat (
224:See also
202:siblings
194:daughter
31:) of an
1191:Catalan
1128:By size
899:-tuples
829:Minimal
732:Regular
623:Mills (
559:Cuban (
435:Proth (
387:classes
305:(ed.).
132:in the
129:A037274
113:base 10
33:integer
1236:Strong
1226:Perrin
1211:Fermat
987:Sexy (
907:Twin (
849:Unique
777:Unique
737:Strong
727:Pillai
707:Wilson
671:Perrin
198:parent
187:) for
122:) are
20:, the
1221:Lucas
1201:Euler
854:Happy
801:Emirp
767:Higgs
762:Super
742:Stern
712:Lucky
656:Lucas
251:Notes
210:lives
151:proof
1144:list
1079:Chen
859:Self
787:Base
757:Good
691:pair
661:Pell
612:−1)·
311:The
214:OEIS
196:and
134:OEIS
105:wiki
1615:281
1610:277
1605:271
1600:269
1595:263
1590:257
1585:251
1580:241
1575:239
1570:233
1565:229
1560:227
1555:223
1550:211
1545:199
1540:197
1535:193
1530:191
1525:181
1520:179
1515:173
1510:167
1505:163
1500:157
1495:151
1490:149
1485:139
1480:137
1475:131
1470:127
1465:113
1460:109
1455:107
1450:103
1445:101
1105:, 2
1089:, 2
1010:a·n
568:)/(
99:'s
72:HPn
64:HPn
50:of
16:In
1641::
1440:97
1435:89
1430:83
1425:79
1420:73
1415:71
1410:67
1405:61
1400:59
1395:53
1390:47
1385:43
1380:41
1375:37
1370:31
1365:29
1360:23
1355:19
1350:17
1345:13
1340:11
1037:,
1033:,
1012:,
992:,
976:,
952:,
928:,
912:,
341:28
339:,
309:.
86:.
25:HP
1335:7
1330:5
1325:3
1320:2
1119:)
1115:p
1111:p
1107:p
1103:p
1095:)
1091:p
1087:p
1074:)
1070:n
1066:n
1062:n
1046:)
1043:n
1039:p
1035:p
1031:n
1027:p
1018:)
1014:n
1006:p
998:)
994:p
990:p
982:)
978:p
974:p
966:)
962:p
958:p
954:p
950:p
942:)
938:p
934:p
930:p
926:p
918:)
914:p
910:p
897:k
693:)
689:(
635:)
632:⌋
629:A
626:⌊
618:)
614:b
610:b
608:(
601:)
592:)
589:y
585:x
577:)
574:y
570:x
566:y
562:x
554:)
550:n
542:)
538:n
530:)
521:)
518:y
514:x
506:)
497:)
493:n
491:4
484:)
479:n
477:p
469:)
464:n
462:p
454:)
450:n
442:)
438:k
422:)
413:)
404:)
377:e
370:t
363:v
288:.
270:.
189:n
185:n
136:)
120:n
68:m
66:(
60:n
56:m
36:n
29:n
27:(
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