332:
1134:
929:
1396:
211:
401:(2) = {3, 5, 11, 13, 19, 29, 37, 53, 59, 61, 67, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 197, 211, 227, 269, 293, 317, 347, 349, 373, 379, 389, 419, 421, 443, 461, 467, 491, ...}.
861:
1235:
794:
533:
645:
564:
982:
1517:
736:
765:
1160:
700:
1431:
594:
674:
1018:
1180:
1061:
1038:
949:
814:
614:
478:
1717:
391:
341:
195:
1712:
1066:
1545:
1510:
1540:
87:
on
September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2024. In fact, there is no single value of
1681:
984:
with complex multiplication under the
Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field.
1597:
1503:
866:
1468:
Hasse, H (1966). "About the density of prime numbers p, for a given integral number a not equal to 0 of even or odd order mod p".
1707:
1535:
1702:
429:
1627:
1265:
1567:
327:{\displaystyle C_{\mathrm {Artin} }=\prod _{p\ \mathrm {prime} }\left(1-{\frac {1}{p(p-1)}}\right)=0.3739558136\ldots }
27:
This article is about the conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see
405:
It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to
1602:
1676:
1552:
819:
1661:
1607:
1255:
1185:
1587:
354:
does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of
443:
proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes
1666:
1634:
1622:
1651:
1592:
1572:
1557:
1252:, a number that plays the same role in a generalization of Artin's conjecture as Artin's constant plays here
54:
1646:
1577:
1639:
1425:
1612:
1249:
770:
483:
1656:
1617:
619:
538:
447:. He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
957:
1671:
124:
705:
1582:
1481:
1329:
1260:
421:
185:
69:
741:
440:
425:
1139:
679:
1473:
1411:
1356:
1313:
28:
1325:
572:
1321:
650:
73:
995:
1286:
1165:
1046:
1023:
934:
799:
599:
463:
1696:
1485:
1333:
174:
50:
35:
17:
1416:
84:
58:
992:
Krishnamurty proposed the question how often the period of the decimal expansion
931:. Here we exclude the primes which divide the denominators of the coordinates of
1378:
1317:
80:
65:
1129:{\displaystyle g^{\left({\frac {p-1}{2^{j}}}\right)}\not \equiv 1{\bmod {p}}}
158:) has a positive asymptotic density inside the set of primes. In particular,
1360:
1495:
1347:
D. R. Heath-Brown (March 1986). "Artin's
Conjecture for Primitive Roots".
1043:
The claim is that the period of the decimal expansion of a prime in base
435:
Without the generalized
Riemann hypothesis, there is no single value of
1477:
72:
to these primes. This conjectural density equals Artin's constant or a
43:
103:
be an integer that is not a square number and not −1. Write
350:
Similar conjectural product formulas exist for the density when
1499:
1446:
924:{\displaystyle N(P)\sim C_{E}\left({\frac {x}{\log x}}\right)}
1117:
535:, Lang and Trotter gave a conjecture for rational points on
954:
Gupta and Murty proved the Lang and
Trotter conjecture for
386:
336:
190:
377:
for which 2 is a primitive root has the above density
1304:
Hooley, Christopher (1967). "On Artin's conjecture".
1188:
1168:
1142:
1069:
1049:
1026:
998:
960:
937:
869:
822:
802:
773:
744:
708:
682:
653:
622:
602:
575:
541:
486:
466:
214:
1229:
1174:
1154:
1128:
1055:
1032:
1012:
976:
943:
923:
855:
808:
788:
759:
730:
694:
668:
639:
608:
588:
558:
527:
472:
428:for the conjecture, assuming certain cases of the
373:= 2. The conjecture claims that the set of primes
326:
569:Specifically, they said there exists a constant
566:analogous to Artin's primitive root conjecture.
206:, which can be expressed as an infinite product
1379:"Artin's Primitive Root Conjecture â a survey"
1511:
8:
856:{\displaystyle {\bar {E}}(\mathbb {F_{p}} )}
412:) is 38/95 = 2/5 = 0.4.
1430:: CS1 maint: numeric names: authors list (
1230:{\displaystyle p\equiv 1+2^{j}\mod {2^{j}}}
1518:
1504:
1496:
1415:
1221:
1216:
1215:
1205:
1187:
1167:
1141:
1120:
1116:
1096:
1079:
1074:
1068:
1048:
1025:
1002:
997:
970:
969:
964:
959:
936:
899:
889:
868:
844:
843:
840:
838:
824:
823:
821:
801:
780:
779:
776:
774:
772:
746:
745:
743:
712:
707:
681:
652:
630:
629:
621:
601:
580:
574:
549:
548:
540:
504:
491:
485:
465:
283:
253:
246:
220:
219:
213:
1240:The result was proven by Hasse in 1966.
439:for which Artin's conjecture is proved.
91:for which Artin's conjecture is proved.
1277:
702:) for which the reduction of the point
1423:
384:. The set of such primes is (sequence
1447:"Primitive points on elliptic curves"
1397:"Primitive points on Elliptic Curves"
767:generates the whole set of points in
40:Artin's conjecture on primitive roots
7:
1372:
1370:
1349:The Quarterly Journal of Mathematics
596:for a given point of infinite order
720:
1718:Unsolved problems in number theory
451:Some variations of Artin's problem
266:
263:
260:
257:
254:
233:
230:
227:
224:
221:
198:), this density is independent of
25:
1285:Michon, Gerard P. (2006-06-15).
789:{\displaystyle \mathbb {F_{p}} }
528:{\displaystyle y^{2}=x^{3}+ax+b}
1713:Conjectures about prime numbers
1417:10.1090/S0002-9904-1977-14310-3
1211:
713:
640:{\displaystyle E(\mathbb {Q} )}
559:{\displaystyle E(\mathbb {Q} )}
977:{\displaystyle E/\mathbb {Q} }
879:
873:
850:
835:
829:
751:
724:
714:
663:
657:
634:
626:
616:in the set of rational points
553:
545:
430:generalized Riemann hypothesis
304:
292:
1:
1182:is unique and p is such that
147:. Then the conjecture states
1266:Cyclic number (group theory)
731:{\displaystyle P{\pmod {p}}}
1395:Lang and 2 Trotter (1977).
1256:BrownâZassenhaus conjecture
143:is a primitive root modulo
135:) the set of prime numbers
79:The conjecture was made by
1734:
760:{\displaystyle {\bar {P}}}
169:Under the conditions that
26:
1531:
1318:10.1515/crll.1967.225.209
1063:is even if and only if
1526:Prime number conjectures
1445:Gupta and Murty (1987).
188:to 1 modulo 4 (sequence
1708:Algebraic number theory
1677:Schinzel's hypothesis H
1155:{\displaystyle j\geq 1}
695:{\displaystyle p\leq x}
450:
57:modulo infinitely many
1703:Analytic number theory
1451:Compositio Mathematica
1231:
1176:
1156:
1130:
1057:
1034:
1014:
978:
945:
925:
857:
810:
790:
761:
732:
696:
670:
641:
610:
590:
560:
529:
474:
328:
1682:Waring's prime number
1470:Mathematische Annalen
1404:Bull. Amer. Math. Soc
1361:10.1093/qmath/37.1.27
1232:
1177:
1157:
1131:
1058:
1035:
1015:
979:
946:
926:
858:
811:
791:
762:
733:
697:
671:
647:such that the number
642:
611:
591:
589:{\displaystyle C_{E}}
561:
530:
475:
329:
18:Artin's constant
1306:J. Reine Angew. Math
1186:
1166:
1140:
1067:
1047:
1024:
996:
958:
935:
867:
820:
800:
771:
742:
706:
680:
669:{\displaystyle N(P)}
651:
620:
600:
573:
539:
484:
464:
212:
42:states that a given
1647:Legendre's constant
1013:{\displaystyle 1/p}
1598:ElliottâHalberstam
1583:Chinese hypothesis
1478:10.1007/BF01361432
1287:"Artin's Constant"
1261:Full reptend prime
1250:Stephens' constant
1227:
1172:
1152:
1126:
1053:
1030:
1010:
974:
941:
921:
853:
806:
786:
757:
728:
692:
666:
637:
606:
586:
556:
525:
470:
460:An elliptic curve
422:Christopher Hooley
369:For example, take
324:
271:
76:multiple thereof.
70:asymptotic density
49:that is neither a
1690:
1689:
1618:Landau's problems
1175:{\displaystyle j}
1102:
1056:{\displaystyle g}
1033:{\displaystyle p}
944:{\displaystyle P}
915:
832:
809:{\displaystyle E}
754:
609:{\displaystyle P}
473:{\displaystyle E}
441:D. R. Heath-Brown
426:conditional proof
308:
252:
242:
68:also ascribes an
16:(Redirected from
1725:
1536:HardyâLittlewood
1520:
1513:
1506:
1497:
1490:
1489:
1465:
1459:
1458:
1442:
1436:
1435:
1429:
1421:
1419:
1401:
1392:
1386:
1385:
1383:
1374:
1365:
1364:
1344:
1338:
1337:
1312:(225): 209â220.
1301:
1295:
1294:
1282:
1236:
1234:
1233:
1228:
1226:
1225:
1210:
1209:
1181:
1179:
1178:
1173:
1161:
1159:
1158:
1153:
1135:
1133:
1132:
1127:
1125:
1124:
1109:
1108:
1107:
1103:
1101:
1100:
1091:
1080:
1062:
1060:
1059:
1054:
1039:
1037:
1036:
1031:
1019:
1017:
1016:
1011:
1006:
983:
981:
980:
975:
973:
968:
950:
948:
947:
942:
930:
928:
927:
922:
920:
916:
914:
900:
894:
893:
862:
860:
859:
854:
849:
848:
847:
834:
833:
825:
815:
813:
812:
807:
795:
793:
792:
787:
785:
784:
783:
766:
764:
763:
758:
756:
755:
747:
737:
735:
734:
729:
727:
701:
699:
698:
693:
675:
673:
672:
667:
646:
644:
643:
638:
633:
615:
613:
612:
607:
595:
593:
592:
587:
585:
584:
565:
563:
562:
557:
552:
534:
532:
531:
526:
509:
508:
496:
495:
479:
477:
476:
471:
389:
339:
333:
331:
330:
325:
314:
310:
309:
307:
284:
270:
269:
250:
238:
237:
236:
204:Artin's constant
193:
29:Artin L-function
21:
1733:
1732:
1728:
1727:
1726:
1724:
1723:
1722:
1693:
1692:
1691:
1686:
1527:
1524:
1494:
1493:
1467:
1466:
1462:
1444:
1443:
1439:
1422:
1399:
1394:
1393:
1389:
1381:
1377:Moree, Pieter.
1376:
1375:
1368:
1346:
1345:
1341:
1303:
1302:
1298:
1284:
1283:
1279:
1274:
1246:
1217:
1201:
1184:
1183:
1164:
1163:
1138:
1137:
1092:
1081:
1075:
1070:
1065:
1064:
1045:
1044:
1022:
1021:
994:
993:
990:
985:
956:
955:
933:
932:
904:
895:
885:
865:
864:
839:
818:
817:
798:
797:
775:
769:
768:
740:
739:
704:
703:
678:
677:
649:
648:
618:
617:
598:
597:
576:
571:
570:
537:
536:
500:
487:
482:
481:
462:
461:
458:
453:
418:
416:Partial results
411:
385:
383:
367:
360:
335:
288:
276:
272:
215:
210:
209:
189:
183:
123:
113:
97:
32:
23:
22:
15:
12:
11:
5:
1731:
1729:
1721:
1720:
1715:
1710:
1705:
1695:
1694:
1688:
1687:
1685:
1684:
1679:
1674:
1669:
1664:
1659:
1654:
1649:
1644:
1643:
1642:
1637:
1632:
1631:
1630:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1575:
1570:
1565:
1560:
1555:
1550:
1549:
1548:
1543:
1532:
1529:
1528:
1525:
1523:
1522:
1515:
1508:
1500:
1492:
1491:
1460:
1437:
1410:(2): 289â292.
1387:
1366:
1339:
1296:
1276:
1275:
1273:
1270:
1269:
1268:
1263:
1258:
1253:
1245:
1242:
1224:
1220:
1214:
1208:
1204:
1200:
1197:
1194:
1191:
1171:
1151:
1148:
1145:
1123:
1119:
1115:
1112:
1106:
1099:
1095:
1090:
1087:
1084:
1078:
1073:
1052:
1029:
1009:
1005:
1001:
989:
986:
972:
967:
963:
953:
940:
919:
913:
910:
907:
903:
898:
892:
888:
884:
881:
878:
875:
872:
863:, is given by
852:
846:
842:
837:
831:
828:
805:
782:
778:
753:
750:
726:
723:
719:
716:
711:
691:
688:
685:
665:
662:
659:
656:
636:
632:
628:
625:
605:
583:
579:
555:
551:
547:
544:
524:
521:
518:
515:
512:
507:
503:
499:
494:
490:
469:
457:
456:Elliptic curve
454:
452:
449:
417:
414:
409:
403:
402:
381:
366:
363:
358:
348:
347:
346:
345:
323:
320:
317:
313:
306:
303:
300:
297:
294:
291:
287:
282:
279:
275:
268:
265:
262:
259:
256:
249:
245:
241:
235:
232:
229:
226:
223:
218:
181:
167:
166:) is infinite.
121:
111:
96:
93:
55:primitive root
24:
14:
13:
10:
9:
6:
4:
3:
2:
1730:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1700:
1698:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1641:
1638:
1636:
1633:
1629:
1626:
1625:
1624:
1621:
1620:
1619:
1616:
1614:
1611:
1609:
1606:
1604:
1603:Firoozbakht's
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1579:
1576:
1574:
1571:
1569:
1566:
1564:
1561:
1559:
1556:
1554:
1551:
1547:
1544:
1542:
1539:
1538:
1537:
1534:
1533:
1530:
1521:
1516:
1514:
1509:
1507:
1502:
1501:
1498:
1487:
1483:
1479:
1475:
1471:
1464:
1461:
1456:
1452:
1448:
1441:
1438:
1433:
1427:
1418:
1413:
1409:
1405:
1398:
1391:
1388:
1380:
1373:
1371:
1367:
1362:
1358:
1354:
1350:
1343:
1340:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1307:
1300:
1297:
1292:
1288:
1281:
1278:
1271:
1267:
1264:
1262:
1259:
1257:
1254:
1251:
1248:
1247:
1243:
1241:
1238:
1222:
1218:
1212:
1206:
1202:
1198:
1195:
1192:
1189:
1169:
1149:
1146:
1143:
1121:
1113:
1110:
1104:
1097:
1093:
1088:
1085:
1082:
1076:
1071:
1050:
1041:
1027:
1007:
1003:
999:
987:
965:
961:
952:
938:
917:
911:
908:
905:
901:
896:
890:
886:
882:
876:
870:
826:
816:, denoted by
803:
748:
721:
717:
709:
689:
686:
683:
660:
654:
623:
603:
581:
577:
567:
542:
522:
519:
516:
513:
510:
505:
501:
497:
492:
488:
467:
455:
448:
446:
442:
438:
433:
431:
427:
423:
415:
413:
408:
400:
397:
396:
395:
393:
388:
380:
376:
372:
364:
362:
357:
353:
343:
338:
321:
318:
315:
311:
301:
298:
295:
289:
285:
280:
277:
273:
247:
243:
239:
216:
208:
207:
205:
201:
197:
192:
187:
180:
176:
175:perfect power
172:
168:
165:
161:
157:
153:
150:
149:
148:
146:
142:
138:
134:
130:
126:
120:
116:
110:
107: =
106:
102:
94:
92:
90:
86:
82:
77:
75:
71:
67:
63:
60:
56:
52:
51:square number
48:
45:
41:
37:
36:number theory
30:
19:
1568:BatemanâHorn
1562:
1469:
1463:
1454:
1450:
1440:
1426:cite journal
1407:
1403:
1390:
1355:(1): 27â38.
1352:
1348:
1342:
1309:
1305:
1299:
1290:
1280:
1239:
1042:
991:
568:
459:
444:
436:
434:
424:published a
419:
406:
404:
398:
378:
374:
370:
368:
355:
351:
349:
319:0.3739558136
203:
199:
178:
170:
163:
159:
155:
151:
144:
140:
136:
132:
128:
127:. Denote by
118:
114:
108:
104:
100:
98:
88:
85:Helmut Hasse
78:
61:
53:nor â1 is a
46:
39:
33:
1662:Oppermann's
1608:Gilbreath's
1578:Bunyakovsky
1020:of a prime
738:denoted by
676:of primes (
202:and equals
125:square-free
95:Formulation
1697:Categories
1667:Polignac's
1640:Twin prime
1635:Legendre's
1623:Goldbach's
1553:AgohâGiuga
1291:Numericana
1272:References
988:Even order
334:(sequence
139:such that
81:Emil Artin
66:conjecture
1652:Lemoine's
1593:Dickson's
1573:Brocard's
1558:Andrica's
1486:121171472
1472:: 19â23.
1334:117943829
1193:≡
1147:≥
1086:−
1040:is even.
909:
883:∼
830:¯
752:¯
687:≤
480:given by
420:In 1967,
322:…
299:−
281:−
244:∏
186:congruent
177:and that
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1588:Cramér's
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1244:See also
1111:≢
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340:in the
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194:in the
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184:is not
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1136:where
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1672:PĂłlya
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410:Artin
382:Artin
359:Artin
117:with
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1432:link
1310:1967
1162:and
392:OEIS
342:OEIS
196:OEIS
99:Let
1546:2nd
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1474:doi
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