1005:, for g sufficiently large (i.e. g>20), then indeed Goupta and Murty obtainded an asymptotic formula for all prime numbers with positive main term, for some particular classes of elliptic curves with complex multiplication, and one could ask what are the minimum subsets of P1, P2,...,Pg, for which such asymptotic formula exists. This result could be considered as an analogous to Artin's primitive root conjecture!
974:
stated for all prime numbers and which probably is totally false! Moreover Gupta and Murty proved that the main term of their asymptotic formula is positive for some particular classes of elliptic curves with complex multiplication. If one consideres the group P generated by several independently linear points P1, P2,...,Pg in
973:
with complex multiplication under the
Generalized Riemann Hypothesis, for primes splitting in the relevant imaginary quadratic field, i. e., they proved some asymptotic formula for only half of the prime numbers without actually verifying any particular case of Lang and Trotter conjecture which is
321:
1157:
918:
1419:
200:
390:(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, ...}.
850:
1258:
783:
522:
1003:
634:
553:
971:
1540:
725:
754:
1183:
689:
1454:
583:
663:
1041:
1203:
1084:
1061:
938:
803:
603:
467:
1740:
380:
330:
184:
1735:
1089:
1568:
1533:
1563:
76:
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
1704:
1620:
1526:
855:
1491:
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".
1730:
1558:
1725:
418:
1650:
1288:
1590:
316:{\displaystyle C_{\mathrm {Artin} }=\prod _{p\ \mathrm {prime} }\left(1-{\frac {1}{p(p-1)}}\right)=0.3739558136\ldots }
16:
This article is about the conjecture of Emil Artin on primitive roots. For the conjecture of Artin on L-functions, see
394:
It has 38 elements smaller than 500 and there are 95 primes smaller than 500. The ratio (which conjecturally tends to
1625:
1699:
1575:
808:
1684:
1630:
1278:
1208:
1610:
343:
does not satisfy the above conditions. In these cases, the conjectural density is always a rational multiple of
432:
proved in 1986 (Corollary 1) that at least one of 2, 3, or 5 is a primitive root modulo infinitely many primes
1689:
1657:
1645:
1674:
1615:
1595:
1580:
1275:, a number that plays the same role in a generalization of Artin's conjecture as Artin's constant plays here
43:
1669:
1600:
1662:
1448:
1635:
1272:
759:
472:
1679:
1640:
977:
608:
527:
436:. He also proved (Corollary 2) that there are at most two primes for which Artin's conjecture fails.
946:
1694:
113:
694:
1605:
1504:
1352:
1283:
410:
174:
58:
730:
429:
414:
1162:
668:
1496:
1434:
1379:
1336:
17:
1348:
561:
1344:
639:
62:
1018:
1309:
1188:
1069:
1046:
923:
788:
588:
452:
1719:
1508:
1356:
163:
39:
24:
1439:
73:
47:
1015:
Krishnamurty proposed the question how often the period of the decimal expansion
920:. Here we exclude the primes which divide the denominators of the coordinates of
1401:
1340:
69:
54:
1152:{\displaystyle g^{\left({\frac {p-1}{2^{j}}}\right)}\not \equiv 1{\bmod {p}}}
147:) has a positive asymptotic density inside the set of primes. In particular,
1383:
1518:
1370:
D. R. Heath-Brown (March 1986). "Artin's
Conjecture for Primitive Roots".
1066:
The claim is that the period of the decimal expansion of a prime in base
424:
Without the generalized
Riemann hypothesis, there is no single value of
1500:
61:
to these primes. This conjectural density equals Artin's constant or a
32:
92:
be an integer that is not a square number and not −1. Write
339:
Similar conjectural product formulas exist for the density when
1522:
1469:
943:
Gupta and Murty proved... the Lang and
Trotter conjecture for
913:{\displaystyle N(P)\sim C_{E}\left({\frac {x}{\log x}}\right)}
1140:
524:, Lang and Trotter gave a conjecture for rational points on
375:
325:
179:
366:
for which 2 is a primitive root has the above density
1327:
Hooley, Christopher (1967). "On Artin's conjecture".
1211:
1191:
1165:
1092:
1072:
1049:
1021:
980:
949:
926:
858:
811:
791:
762:
733:
697:
671:
642:
611:
591:
564:
530:
475:
455:
203:
1252:
1197:
1177:
1151:
1078:
1055:
1035:
997:
965:
932:
912:
844:
797:
777:
748:
719:
683:
657:
628:
597:
577:
547:
516:
461:
417:for the conjecture, assuming certain cases of the
362:= 2. The conjecture claims that the set of primes
315:
558:Specifically, they said there exists a constant
555:analogous to Artin's primitive root conjecture.
195:, which can be expressed as an infinite product
1402:"Artin's Primitive Root Conjecture â a survey"
1534:
8:
845:{\displaystyle {\bar {E}}(\mathbb {F_{p}} )}
401:) is 38/95 = 2/5 = 0.4.
1453:: CS1 maint: numeric names: authors list (
1253:{\displaystyle p\equiv 1+2^{j}\mod {2^{j}}}
1541:
1527:
1519:
1438:
1244:
1239:
1238:
1228:
1210:
1190:
1164:
1143:
1139:
1119:
1102:
1097:
1091:
1071:
1048:
1025:
1020:
988:
987:
979:
959:
958:
953:
948:
925:
888:
878:
857:
833:
832:
829:
827:
813:
812:
810:
790:
769:
768:
765:
763:
761:
735:
734:
732:
701:
696:
670:
641:
619:
618:
610:
590:
569:
563:
538:
537:
529:
493:
480:
474:
454:
272:
242:
235:
209:
208:
202:
1263:The result was proven by Hasse in 1966.
428:for which Artin's conjecture is proved.
80:for which Artin's conjecture is proved.
1300:
691:) for which the reduction of the point
1446:
373:. The set of such primes is (sequence
1470:"Primitive points on elliptic curves"
1420:"Primitive points on Elliptic Curves"
756:generates the whole set of points in
29:Artin's conjecture on primitive roots
7:
1395:
1393:
1372:The Quarterly Journal of Mathematics
585:for a given point of infinite order
709:
1741:Unsolved problems in number theory
440:Some variations of Artin's problem
255:
252:
249:
246:
243:
222:
219:
216:
213:
210:
187:), this density is independent of
14:
1308:Michon, Gerard P. (2006-06-15).
778:{\displaystyle \mathbb {F_{p}} }
517:{\displaystyle y^{2}=x^{3}+ax+b}
1736:Conjectures about prime numbers
1440:10.1090/S0002-9904-1977-14310-3
1234:
998:{\displaystyle E(\mathbb {Q} )}
702:
629:{\displaystyle E(\mathbb {Q} )}
548:{\displaystyle E(\mathbb {Q} )}
992:
984:
966:{\displaystyle E/\mathbb {Q} }
868:
862:
839:
824:
818:
740:
713:
703:
652:
646:
623:
615:
605:in the set of rational points
542:
534:
419:generalized Riemann hypothesis
293:
281:
1:
1205:is unique and p is such that
136:. Then the conjecture states
1289:Cyclic number (group theory)
720:{\displaystyle P{\pmod {p}}}
1418:Lang and 2 Trotter (1977).
1279:BrownâZassenhaus conjecture
132:is a primitive root modulo
124:) the set of prime numbers
68:The conjecture was made by
1757:
749:{\displaystyle {\bar {P}}}
158:Under the conditions that
15:
1554:
1341:10.1515/crll.1967.225.209
1086:is even if and only if
1549:Prime number conjectures
1468:Gupta and Murty (1987).
177:to 1 modulo 4 (sequence
1731:Algebraic number theory
1700:Schinzel's hypothesis H
1178:{\displaystyle j\geq 1}
684:{\displaystyle p\leq x}
439:
46:modulo infinitely many
1726:Analytic number theory
1474:Compositio Mathematica
1254:
1199:
1179:
1153:
1080:
1057:
1037:
999:
967:
934:
914:
846:
799:
779:
750:
721:
685:
659:
630:
599:
579:
549:
518:
463:
317:
1705:Waring's prime number
1493:Mathematische Annalen
1427:Bull. Amer. Math. Soc
1384:10.1093/qmath/37.1.27
1255:
1200:
1180:
1154:
1081:
1058:
1038:
1000:
968:
935:
915:
847:
800:
780:
751:
722:
686:
660:
636:such that the number
631:
600:
580:
578:{\displaystyle C_{E}}
550:
519:
464:
318:
1329:J. Reine Angew. Math
1209:
1189:
1163:
1090:
1070:
1047:
1019:
978:
947:
924:
856:
809:
789:
760:
731:
695:
669:
658:{\displaystyle N(P)}
640:
609:
589:
562:
528:
473:
453:
201:
31:states that a given
1670:Legendre's constant
1036:{\displaystyle 1/p}
1621:ElliottâHalberstam
1606:Chinese hypothesis
1501:10.1007/BF01361432
1310:"Artin's Constant"
1284:Full reptend prime
1273:Stephens' constant
1250:
1195:
1175:
1149:
1076:
1053:
1033:
995:
963:
930:
910:
842:
795:
775:
746:
717:
681:
655:
626:
595:
575:
545:
514:
459:
449:An elliptic curve
411:Christopher Hooley
358:For example, take
313:
260:
65:multiple thereof.
59:asymptotic density
38:that is neither a
1713:
1712:
1641:Landau's problems
1198:{\displaystyle j}
1125:
1079:{\displaystyle g}
1056:{\displaystyle p}
933:{\displaystyle P}
904:
821:
798:{\displaystyle E}
743:
598:{\displaystyle P}
462:{\displaystyle E}
430:D. R. Heath-Brown
415:conditional proof
297:
241:
231:
57:also ascribes an
1748:
1559:HardyâLittlewood
1543:
1536:
1529:
1520:
1513:
1512:
1488:
1482:
1481:
1465:
1459:
1458:
1452:
1444:
1442:
1424:
1415:
1409:
1408:
1406:
1397:
1388:
1387:
1367:
1361:
1360:
1335:(225): 209â220.
1324:
1318:
1317:
1305:
1259:
1257:
1256:
1251:
1249:
1248:
1233:
1232:
1204:
1202:
1201:
1196:
1184:
1182:
1181:
1176:
1158:
1156:
1155:
1150:
1148:
1147:
1132:
1131:
1130:
1126:
1124:
1123:
1114:
1103:
1085:
1083:
1082:
1077:
1062:
1060:
1059:
1054:
1042:
1040:
1039:
1034:
1029:
1004:
1002:
1001:
996:
991:
972:
970:
969:
964:
962:
957:
939:
937:
936:
931:
919:
917:
916:
911:
909:
905:
903:
889:
883:
882:
851:
849:
848:
843:
838:
837:
836:
823:
822:
814:
804:
802:
801:
796:
784:
782:
781:
776:
774:
773:
772:
755:
753:
752:
747:
745:
744:
736:
726:
724:
723:
718:
716:
690:
688:
687:
682:
664:
662:
661:
656:
635:
633:
632:
627:
622:
604:
602:
601:
596:
584:
582:
581:
576:
574:
573:
554:
552:
551:
546:
541:
523:
521:
520:
515:
498:
497:
485:
484:
468:
466:
465:
460:
378:
328:
322:
320:
319:
314:
303:
299:
298:
296:
273:
259:
258:
239:
227:
226:
225:
193:Artin's constant
182:
18:Artin L-function
1756:
1755:
1751:
1750:
1749:
1747:
1746:
1745:
1716:
1715:
1714:
1709:
1550:
1547:
1517:
1516:
1490:
1489:
1485:
1467:
1466:
1462:
1445:
1422:
1417:
1416:
1412:
1404:
1400:Moree, Pieter.
1399:
1398:
1391:
1369:
1368:
1364:
1326:
1325:
1321:
1307:
1306:
1302:
1297:
1269:
1240:
1224:
1207:
1206:
1187:
1186:
1161:
1160:
1115:
1104:
1098:
1093:
1088:
1087:
1068:
1067:
1045:
1044:
1017:
1016:
1013:
1006:
976:
975:
945:
944:
922:
921:
893:
884:
874:
854:
853:
828:
807:
806:
787:
786:
764:
758:
757:
729:
728:
693:
692:
667:
666:
638:
637:
607:
606:
587:
586:
565:
560:
559:
526:
525:
489:
476:
471:
470:
451:
450:
447:
442:
407:
405:Partial results
400:
374:
372:
356:
349:
324:
277:
265:
261:
204:
199:
198:
178:
172:
112:
102:
86:
21:
12:
11:
5:
1754:
1752:
1744:
1743:
1738:
1733:
1728:
1718:
1717:
1711:
1710:
1708:
1707:
1702:
1697:
1692:
1687:
1682:
1677:
1672:
1667:
1666:
1665:
1660:
1655:
1654:
1653:
1638:
1633:
1628:
1623:
1618:
1613:
1608:
1603:
1598:
1593:
1588:
1583:
1578:
1573:
1572:
1571:
1566:
1555:
1552:
1551:
1548:
1546:
1545:
1538:
1531:
1523:
1515:
1514:
1483:
1460:
1433:(2): 289â292.
1410:
1389:
1362:
1319:
1299:
1298:
1296:
1293:
1292:
1291:
1286:
1281:
1276:
1268:
1265:
1247:
1243:
1237:
1231:
1227:
1223:
1220:
1217:
1214:
1194:
1174:
1171:
1168:
1146:
1142:
1138:
1135:
1129:
1122:
1118:
1113:
1110:
1107:
1101:
1096:
1075:
1052:
1032:
1028:
1024:
1012:
1009:
994:
990:
986:
983:
961:
956:
952:
942:
929:
908:
902:
899:
896:
892:
887:
881:
877:
873:
870:
867:
864:
861:
852:, is given by
841:
835:
831:
826:
820:
817:
794:
771:
767:
742:
739:
715:
712:
708:
705:
700:
680:
677:
674:
654:
651:
648:
645:
625:
621:
617:
614:
594:
572:
568:
544:
540:
536:
533:
513:
510:
507:
504:
501:
496:
492:
488:
483:
479:
458:
446:
445:Elliptic curve
443:
441:
438:
406:
403:
398:
392:
391:
370:
355:
352:
347:
337:
336:
335:
334:
312:
309:
306:
302:
295:
292:
289:
286:
283:
280:
276:
271:
268:
264:
257:
254:
251:
248:
245:
238:
234:
230:
224:
221:
218:
215:
212:
207:
170:
156:
155:) is infinite.
110:
100:
85:
82:
44:primitive root
13:
10:
9:
6:
4:
3:
2:
1753:
1742:
1739:
1737:
1734:
1732:
1729:
1727:
1724:
1723:
1721:
1706:
1703:
1701:
1698:
1696:
1693:
1691:
1688:
1686:
1683:
1681:
1678:
1676:
1673:
1671:
1668:
1664:
1661:
1659:
1656:
1652:
1649:
1648:
1647:
1644:
1643:
1642:
1639:
1637:
1634:
1632:
1629:
1627:
1626:Firoozbakht's
1624:
1622:
1619:
1617:
1614:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1577:
1574:
1570:
1567:
1565:
1562:
1561:
1560:
1557:
1556:
1553:
1544:
1539:
1537:
1532:
1530:
1525:
1524:
1521:
1510:
1506:
1502:
1498:
1494:
1487:
1484:
1479:
1475:
1471:
1464:
1461:
1456:
1450:
1441:
1436:
1432:
1428:
1421:
1414:
1411:
1403:
1396:
1394:
1390:
1385:
1381:
1377:
1373:
1366:
1363:
1358:
1354:
1350:
1346:
1342:
1338:
1334:
1330:
1323:
1320:
1315:
1311:
1304:
1301:
1294:
1290:
1287:
1285:
1282:
1280:
1277:
1274:
1271:
1270:
1266:
1264:
1261:
1245:
1241:
1235:
1229:
1225:
1221:
1218:
1215:
1212:
1192:
1172:
1169:
1166:
1144:
1136:
1133:
1127:
1120:
1116:
1111:
1108:
1105:
1099:
1094:
1073:
1064:
1050:
1030:
1026:
1022:
1010:
1008:
981:
954:
950:
941:
927:
906:
900:
897:
894:
890:
885:
879:
875:
871:
865:
859:
815:
805:, denoted by
792:
737:
710:
706:
698:
678:
675:
672:
649:
643:
612:
592:
570:
566:
556:
531:
511:
508:
505:
502:
499:
494:
490:
486:
481:
477:
456:
444:
437:
435:
431:
427:
422:
420:
416:
412:
404:
402:
397:
389:
386:
385:
384:
382:
377:
369:
365:
361:
353:
351:
346:
342:
332:
327:
310:
307:
304:
300:
290:
287:
284:
278:
274:
269:
266:
262:
236:
232:
228:
205:
197:
196:
194:
190:
186:
181:
176:
169:
165:
164:perfect power
161:
157:
154:
150:
146:
142:
139:
138:
137:
135:
131:
127:
123:
119:
115:
109:
105:
99:
96: =
95:
91:
83:
81:
79:
75:
71:
66:
64:
60:
56:
52:
49:
45:
41:
40:square number
37:
34:
30:
26:
25:number theory
19:
1591:BatemanâHorn
1585:
1492:
1486:
1477:
1473:
1463:
1449:cite journal
1430:
1426:
1413:
1378:(1): 27â38.
1375:
1371:
1365:
1332:
1328:
1322:
1313:
1303:
1262:
1065:
1014:
1007:
557:
448:
433:
425:
423:
413:published a
408:
395:
393:
387:
367:
363:
359:
357:
344:
340:
338:
308:0.3739558136
192:
188:
167:
159:
152:
148:
144:
140:
133:
129:
125:
121:
117:
116:. Denote by
107:
103:
97:
93:
89:
87:
77:
74:Helmut Hasse
67:
50:
42:nor â1 is a
35:
28:
22:
1685:Oppermann's
1631:Gilbreath's
1601:Bunyakovsky
1043:of a prime
727:denoted by
665:of primes (
191:and equals
114:square-free
84:Formulation
1720:Categories
1690:Polignac's
1663:Twin prime
1658:Legendre's
1646:Goldbach's
1576:AgohâGiuga
1314:Numericana
1295:References
1011:Even order
323:(sequence
128:such that
70:Emil Artin
55:conjecture
1675:Lemoine's
1616:Dickson's
1596:Brocard's
1581:Andrica's
1509:121171472
1495:: 19â23.
1357:117943829
1216:≡
1170:≥
1109:−
1063:is even.
898:
872:∼
819:¯
741:¯
676:≤
469:given by
409:In 1967,
311:…
288:−
270:−
233:∏
175:congruent
166:and that
162:is not a
1680:Mersenne
1611:Cramér's
1480:: 13â44.
1267:See also
1134:≢
63:rational
1636:Grimm's
1586:Artin's
1349:0207630
379:in the
376:A001122
354:Example
329:in the
326:A005596
183:in the
180:A085397
173:is not
33:integer
1507:
1355:
1347:
1159:where
240:
53:. The
48:primes
1695:PĂłlya
1505:S2CID
1423:(PDF)
1405:(PDF)
1353:S2CID
399:Artin
371:Artin
348:Artin
106:with
1651:weak
1455:link
1333:1967
1185:and
381:OEIS
331:OEIS
185:OEIS
88:Let
1569:2nd
1564:1st
1497:doi
1435:doi
1380:doi
1337:doi
1236:mod
1141:mod
895:log
785:in
707:mod
72:to
23:In
1722::
1503:.
1478:58
1476:.
1472:.
1451:}}
1447:{{
1431:83
1429:.
1425:.
1392:^
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1374:.
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333:).
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1407:.
1386:.
1382::
1359:.
1339::
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1246:j
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1222:+
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1173:1
1167:j
1145:p
1137:1
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1121:j
1117:2
1112:1
1106:p
1100:(
1095:g
1074:g
1051:p
1031:p
1027:/
1023:1
993:)
989:Q
985:(
982:E
960:Q
955:/
951:E
928:P
907:)
901:x
891:x
886:(
880:E
876:C
869:)
866:P
863:(
860:N
840:)
834:p
830:F
825:(
816:E
793:E
770:p
766:F
738:P
714:)
711:p
704:(
699:P
679:x
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653:)
650:P
647:(
644:N
624:)
620:Q
616:(
613:E
593:P
571:E
567:C
543:)
539:Q
535:(
532:E
512:b
509:+
506:x
503:a
500:+
495:3
491:x
487:=
482:2
478:y
457:E
434:p
426:a
396:C
388:S
368:C
364:p
360:a
345:C
341:a
305:=
301:)
294:)
291:1
285:p
282:(
279:p
275:1
267:1
263:(
256:e
253:m
250:i
247:r
244:p
237:p
229:=
223:n
220:i
217:t
214:r
211:A
206:C
189:a
171:0
168:a
160:a
153:a
151:(
149:S
145:a
143:(
141:S
134:p
130:a
126:p
122:a
120:(
118:S
111:0
108:a
104:b
101:0
98:a
94:a
90:a
78:a
51:p
36:a
20:.
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