202:
if all of its left extensions are composite i.e. there is no other left-truncatable prime of which this prime is the left-truncated "tail". Thus 7937 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7937 are all composite, whereas 3797 is a left-truncatable prime that
153:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733,
131:
2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193, 31379, 37337, 37339, 37397, 59393, 59399,
109:
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ... (sequence
66:
is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-right-truncatable prime, since 1825711, 82571, 257, and 5 are all prime.
132:
71933, 73331, 73939, 233993, 239933, 293999, 373379, 373393, 593933, 593993, 719333, 739391, 739393, 739397, 739399, 2339933, 2399333, 2939999, 3733799, 5939333, 7393913, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence
172:
There are 588,939,451 left-and-right-truncatable primes with an even number of digits. The largest is the 104-digit prime 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973.
168:
There are 331,780,864 left-and-right-truncatable primes with an odd number of digits. The largest is the 97-digit prime 7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177.
210:
2, 5, 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, 15647, 16823, 24373, 33547, 34337, 37643, 56983, 57853, 59743, 62383, 63347, 63617, 69337, 72467, 72617, 75653, 76367, 87643, 92683, 97883, 98317, ... (sequence
229:
53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence
253:
used, truncatable primes are defined only in relation with a given base. A variation involves removing 2 or more decimal digits at a time. This is mathematically equivalent to using base 100 or a larger
82:(which started its run in 1968) considered a topic close to that of right-truncatable primes, calling sequences that by adding digits to the right in sequence to an initial number not necessarily prime
154:
739, 751, 757, 773, 821, 823, 827, 829, 839, 853, 857, 859, 877, 929, 937, 953, 971, 977, 1117, 1171, 1193, 1231, 1237, 1291, 1297, 1319, 1373, 1433, 1439, 1471, 1531, 1597, 1613, 1619, ... (sequence
59:
is a prime which remains prime when the last ("right") digit is successively removed. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime.
48:, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime.
146:
The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit.
1189:
421:
792:
225:
Similarly, a right-truncatable prime is called restricted if all of its right extensions are composite. There are 27 restricted right-truncatable primes:
1690:
297:
237:
218:
191:
161:
139:
117:
874:
797:
711:
70:
In base 10, there are exactly 4260 left-truncatable primes, 83 right-truncatable primes, and 920,720,315 left-and-right-truncatable primes.
79:
414:
1695:
1048:
258:, with the restriction that base 10 digits must be at least 10, in order to match a decimal n-digit number with no leading 0.
1129:
407:
1251:
909:
1276:
742:
1184:
817:
1334:
463:
1671:
1261:
914:
822:
1241:
337:
90:
331:
1236:
894:
1344:
1281:
1271:
1256:
889:
747:
384:
176:
There are 15 primes which are both left-truncatable and right-truncatable. They have been called
668:
1313:
1288:
1266:
1246:
841:
534:
313:
1223:
1213:
1208:
1145:
992:
859:
762:
267:
924:
884:
767:
732:
696:
651:
504:
492:
1329:
1303:
1200:
1068:
919:
879:
864:
736:
627:
592:
547:
472:
454:
250:
17:
316:
1684:
1339:
1104:
968:
941:
777:
642:
580:
571:
556:
519:
445:
33:
1660:
1655:
1650:
1645:
1640:
1635:
1630:
1625:
1620:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1520:
1515:
1510:
1505:
1500:
1495:
1490:
1293:
1016:
899:
782:
772:
757:
752:
716:
430:
354:
350:
41:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1231:
904:
812:
807:
787:
701:
604:
480:
380:
343:
287:
255:
1308:
1124:
1032:
952:
802:
706:
369:
184:
2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence
1349:
1298:
1179:
321:
851:
399:
49:
846:
832:
45:
89:
Discussion of the topic dates to at least
November 1969 issue of
52:
representation is often assumed and always used in this article.
403:
291:
232:
213:
186:
156:
134:
112:
149:
There are 920,720,315 left-and-right-truncatable primes:
127:
There are 83 right-truncatable primes. The complete list:
1380:
1375:
1370:
1365:
249:
While the primality of a number does not depend on the
78:
An author named Leslie E. Card in early volumes of the
124:
The largest is the 24-digit 357686312646216567629137.
97:
by two co-authors (Murray Berg and John E. Walstrom).
1358:
1322:
1222:
1199:
1173:
940:
933:
831:
725:
689:
438:
206:There are 1442 restricted left-truncatable primes:
1062: = 0, 1, 2, 3, ...
351:Problems & Puzzles: Puzzle 2.- Prime strings
203:is not restricted because 33797 is also prime.
415:
8:
937:
422:
408:
400:
298:On-Line Encyclopedia of Integer Sequences
105:There are 4260 left-truncatable primes:
279:
93:, where truncatable primes were called
7:
198:A left-truncatable prime is called
80:Journal of Recreational Mathematics
25:
1691:Base-dependent integer sequences
798:Supersingular (moonshine theory)
64:left-and-right-truncatable prime
387:from the original on 2021-12-21
793:Supersingular (elliptic curve)
1:
574:2 ± 2 ± 1
355:Puzzle 131.- Growing primes
1712:
370:"357686312646216567629137"
288:Sloane, N. J. A.
101:Decimal truncatable primes
1669:
1696:Classes of prime numbers
1180:Mega (1,000,000+ digits)
1049:Arithmetic progression (
339:right-truncatable primes
292:"Sequence A077390"
57:right-truncatable prime
18:Right-truncatable prime
1335:Industrial-grade prime
712:Newman–Shanks–Williams
333:left-truncatable prime
38:left-truncatable prime
1672:List of prime numbers
1130:Sophie Germain/Safe (
180:. The complete list:
854:(10 − 1)/9
91:Mathematics Magazine
1163: ± 7, ...
690:By integer sequence
475:(2 + 1)/3
317:"Truncatable Prime"
1345:Formula for primes
978: + 2 or
910:Smarandache–Wellin
368:Grime, Dr. James.
314:Weisstein, Eric W.
301:. OEIS Foundation.
44:which, in a given
1678:
1677:
1289:Carmichael number
1224:Composite numbers
1159: ± 3, 8
1155: ± 1, 4
1118: ± 1, …
1114: ± 1, 4
1110: ± 1, 2
1100:
1099:
645:3·2 − 1
550:2·3 + 1
464:Double Mersenne (
330:Caldwell, Chris,
16:(Redirected from
1703:
1209:Eisenstein prime
1164:
1140:
1119:
1091:
1063:
1043:
1027:
1011:
1006: + 6,
1002: + 2,
987:
982: + 4,
963:
938:
855:
818:Highly cototient
680:
679:
673:
663:
646:
637:
622:
599:
598:·2 − 1
587:
586:·2 + 1
575:
566:
551:
542:
529:
514:
499:
487:
486:·2 + 1
476:
467:
458:
449:
424:
417:
410:
401:
396:
394:
392:
374:
349:Rivera, Carlos,
327:
326:
303:
302:
284:
268:Permutable prime
235:
216:
189:
178:two-sided primes
159:
137:
115:
21:
1711:
1710:
1706:
1705:
1704:
1702:
1701:
1700:
1681:
1680:
1679:
1674:
1665:
1359:First 60 primes
1354:
1318:
1218:
1201:Complex numbers
1195:
1169:
1147:
1131:
1106:
1105:Bi-twin chain (
1096:
1070:
1050:
1034:
1018:
994:
970:
954:
929:
915:Strobogrammatic
853:
827:
721:
685:
677:
671:
670:
653:
644:
629:
606:
594:
582:
573:
558:
549:
536:
528:# + 1
526:
521:
513:# ± 1
511:
506:
498:! ± 1
494:
482:
474:
466:2 − 1
465:
457:2 − 1
456:
448:2 + 1
447:
434:
428:
390:
388:
372:
367:
364:
359:
312:
311:
307:
306:
286:
285:
281:
276:
264:
247:
231:
212:
185:
155:
133:
111:
103:
84:snowball primes
76:
30:
23:
22:
15:
12:
11:
5:
1709:
1707:
1699:
1698:
1693:
1683:
1682:
1676:
1675:
1670:
1667:
1666:
1664:
1663:
1658:
1653:
1648:
1643:
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1583:
1578:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1538:
1533:
1528:
1523:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1483:
1478:
1473:
1468:
1463:
1458:
1453:
1448:
1443:
1438:
1433:
1428:
1423:
1418:
1413:
1408:
1403:
1398:
1393:
1388:
1383:
1378:
1373:
1368:
1362:
1360:
1356:
1355:
1353:
1352:
1347:
1342:
1337:
1332:
1330:Probable prime
1326:
1324:
1323:Related topics
1320:
1319:
1317:
1316:
1311:
1306:
1304:Sphenic number
1301:
1296:
1291:
1286:
1285:
1284:
1279:
1274:
1269:
1264:
1259:
1254:
1249:
1244:
1239:
1228:
1226:
1220:
1219:
1217:
1216:
1214:Gaussian prime
1211:
1205:
1203:
1197:
1196:
1194:
1193:
1192:
1182:
1177:
1175:
1171:
1170:
1168:
1167:
1143:
1139: + 1
1127:
1122:
1101:
1098:
1097:
1095:
1094:
1066:
1046:
1042: + 6
1030:
1026: + 4
1014:
1010: + 8
990:
986: + 6
966:
962: + 2
949:
947:
935:
931:
930:
928:
927:
922:
917:
912:
907:
902:
897:
892:
887:
882:
877:
872:
867:
862:
857:
849:
844:
838:
836:
829:
828:
826:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
729:
727:
723:
722:
720:
719:
714:
709:
704:
699:
693:
691:
687:
686:
684:
683:
666:
662: − 1
649:
640:
625:
602:
590:
578:
569:
554:
545:
541: + 1
532:
524:
517:
509:
502:
490:
478:
470:
461:
452:
442:
440:
436:
435:
429:
427:
426:
419:
412:
404:
398:
397:
363:
362:External links
360:
358:
357:
347:
328:
308:
305:
304:
278:
277:
275:
272:
271:
270:
263:
260:
251:numeral system
246:
243:
242:
241:
223:
222:
196:
195:
166:
165:
144:
143:
122:
121:
102:
99:
75:
72:
29:Type of number
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1708:
1697:
1694:
1692:
1689:
1688:
1686:
1673:
1668:
1662:
1659:
1657:
1654:
1652:
1649:
1647:
1644:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1542:
1539:
1537:
1534:
1532:
1529:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1487:
1484:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1397:
1394:
1392:
1389:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1363:
1361:
1357:
1351:
1348:
1346:
1343:
1341:
1340:Illegal prime
1338:
1336:
1333:
1331:
1328:
1327:
1325:
1321:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1243:
1240:
1238:
1235:
1234:
1233:
1230:
1229:
1227:
1225:
1221:
1215:
1212:
1210:
1207:
1206:
1204:
1202:
1198:
1191:
1188:
1187:
1186:
1185:Largest known
1183:
1181:
1178:
1176:
1172:
1166:
1162:
1158:
1154:
1150:
1144:
1142:
1138:
1134:
1128:
1126:
1123:
1121:
1117:
1113:
1109:
1103:
1102:
1093:
1090:
1087: +
1086:
1082:
1078:
1075: −
1074:
1067:
1065:
1061:
1057:
1054: +
1053:
1047:
1045:
1041:
1037:
1031:
1029:
1025:
1021:
1015:
1013:
1009:
1005:
1001:
997:
991:
989:
985:
981:
977:
973:
967:
965:
961:
957:
951:
950:
948:
946:
944:
939:
936:
932:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
850:
848:
845:
843:
840:
839:
837:
834:
830:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
738:
734:
731:
730:
728:
724:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
694:
692:
688:
682:
676:
667:
665:
661:
657:
650:
648:
641:
639:
636:
633: +
632:
626:
624:
621:
618: −
617:
613:
610: −
609:
603:
601:
597:
591:
589:
585:
579:
577:
570:
568:
565:
562: +
561:
555:
553:
546:
544:
540:
535:Pythagorean (
533:
531:
527:
518:
516:
512:
503:
501:
497:
491:
489:
485:
479:
477:
471:
469:
462:
460:
453:
451:
444:
443:
441:
437:
432:
425:
420:
418:
413:
411:
406:
405:
402:
386:
382:
378:
371:
366:
365:
361:
356:
352:
348:
345:
341:
340:
335:
334:
329:
324:
323:
318:
315:
310:
309:
300:
299:
293:
289:
283:
280:
273:
269:
266:
265:
261:
259:
257:
252:
244:
239:
234:
228:
227:
226:
220:
215:
209:
208:
207:
204:
201:
193:
188:
183:
182:
181:
179:
174:
170:
163:
158:
152:
151:
150:
147:
141:
136:
130:
129:
128:
125:
119:
114:
108:
107:
106:
100:
98:
96:
92:
87:
85:
81:
73:
71:
68:
65:
60:
58:
53:
51:
47:
43:
39:
35:
34:number theory
27:
19:
1294:Almost prime
1252:Euler–Jacobi
1160:
1156:
1152:
1148:
1146:Cunningham (
1136:
1132:
1115:
1111:
1107:
1088:
1084:
1080:
1076:
1072:
1071:consecutive
1059:
1055:
1051:
1039:
1035:
1023:
1019:
1007:
1003:
999:
995:
993:Quadruplet (
983:
979:
975:
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42:prime number
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1232:Pseudoprime
870:Truncatable
842:Palindromic
726:By property
505:Primorial (
493:Factorial (
381:Brady Haran
344:Prime Pages
256:power of 10
245:Other bases
1685:Categories
1314:Pernicious
1309:Interprime
1069:Balanced (
860:Permutable
835:-dependent
652:Williams (
548:Pierpont (
473:Wagstaff
455:Mersenne (
439:By formula
274:References
200:restricted
1350:Prime gap
1299:Semiprime
1262:Frobenius
969:Triplet (
768:Ramanujan
763:Fortunate
733:Wieferich
697:Fibonacci
628:Leyland (
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572:Solinas (
557:Quartan (
346:glossary.
342:, at the
322:MathWorld
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925:Tetradic
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885:Primeval
880:Delicate
865:Circular
852:Repunit
643:Thabit (
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520:Euclid (
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385:Archived
262:See also
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1174:By size
945:-tuples
875:Minimal
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605:Cuban (
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433:classes
391:27 July
377:YouTube
373:(video)
290:(ed.).
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895:Unique
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