980:
861:
270:
577:, and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996). Laerte Sorini, finally, in a work of 2001 showed that a possible counterexample should be a number
1166:
364:
1075:
742:
122:
661:
437:
544:
490:
1361:
1551:
581:
greater than 10 which represents the limit suggested by
Bedocchi for the demonstration technique specified by Giuga to his own conjecture.
1546:
1406:
872:
753:
1389:
1354:
1384:
161:
1525:
1441:
1347:
1086:
281:
995:
1379:
672:
1250:
1471:
1258:
1411:
1446:
1520:
378:
63:
1505:
1451:
603:
1431:
387:
1510:
1478:
1466:
495:
1209:
1495:
1436:
1416:
1401:
1490:
1421:
1267:
1483:
1456:
445:
1500:
1461:
1272:
148:
132:
1515:
590:
144:
128:
1426:
1285:
985:
So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number
1328:
1187:
570:
1293:
1277:
1242:
1226:
1218:
1195:
562:
36:
1297:
1230:
1199:
1319:
Sorini, Laerte (2001). "Un Metodo
Euristico per la Soluzione della Congettura di Giuga".
1178:
Giuga, Giuseppe (1951). "Su una presumibile proprietà caratteristica dei numeri primi".
55:
553:
The statement is still a conjecture since it has not yet been proven that if a number
1540:
1246:
1238:
28:
17:
574:
52:
1321:
Quaderni di
Economia, Matematica e Statistica, DESP, Università di Urbino Carlo Bo
1301:
975:{\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv (-1)^{p-1}\equiv 1{\pmod {p}}.}
856:{\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv (-1)^{p-1}\equiv 1{\pmod {p}},}
1332:
1191:
565:), then the formula does not hold. It has been shown that a composite number
1339:
593:, which has been proven to be true. Wilson's theorem states that a number
1289:
1222:
373:
being prime is sufficient for the second equivalence to hold, since if
265:{\displaystyle 1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}\equiv -1{\pmod {p}}}
1281:
1343:
1161:{\displaystyle \prod _{i=1}^{p-1}i^{p-1}\equiv 1{\pmod {p}}.}
359:{\displaystyle \sum _{i=1}^{p-1}i^{p-1}\equiv -1{\pmod {p}}.}
1070:{\displaystyle \sum _{i=1}^{p-1}i^{p-1}\equiv -1{\pmod {p}}}
737:{\displaystyle \prod _{i=1}^{p-1}i\equiv -1{\pmod {p}}.}
1089:
998:
875:
756:
675:
606:
498:
448:
390:
284:
164:
66:
1180:
Ist.Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur.
1160:
1069:
974:
855:
736:
655:
569:satisfies the formula if and only if it is both a
538:
484:
431:
358:
264:
116:
589:The Agoh–Giuga conjecture bears a similarity to
1207:Agoh, Takashi (1995). "On Giuga's conjecture".
1355:
117:{\displaystyle pB_{p-1}\equiv -1{\pmod {p}}.}
8:
147:(1990); an equivalent formulation is due to
656:{\displaystyle (p-1)!\equiv -1{\pmod {p}},}
1362:
1348:
1340:
432:{\displaystyle a^{p-1}\equiv 1{\pmod {p}}}
1271:
1139:
1121:
1105:
1094:
1088:
1051:
1030:
1014:
1003:
997:
953:
935:
907:
891:
880:
874:
834:
816:
788:
772:
761:
755:
715:
691:
680:
674:
634:
605:
517:
497:
447:
413:
395:
389:
337:
316:
300:
289:
283:
246:
225:
188:
169:
163:
143:The conjecture as stated above is due to
95:
74:
65:
539:{\displaystyle p-1\equiv -1{\pmod {p}}.}
492:, and the equivalence follows, since
7:
1147:
1059:
961:
842:
723:
642:
525:
421:
345:
254:
103:
1552:Unsolved problems in number theory
25:
1251:"Giuga's Conjecture on Primality"
485:{\displaystyle a=1,2,\dots ,p-1}
151:, from 1950, to the effect that
1547:Conjectures about prime numbers
1140:
1052:
954:
835:
716:
635:
518:
414:
338:
247:
96:
1151:
1141:
1063:
1053:
965:
955:
932:
922:
846:
836:
813:
803:
727:
717:
646:
636:
619:
607:
529:
519:
425:
415:
349:
339:
258:
248:
222:
209:
107:
97:
1:
1259:American Mathematical Monthly
666:which may also be written as
275:which may also be written as
585:Relation to Wilson's theorem
747:For an odd prime p we have
369:It is trivial to show that
1568:
1375:
1249:; Girgensohn, R. (1996).
1370:Prime number conjectures
989:is prime if and only if
597:is prime if and only if
155:is prime if and only if
1521:Schinzel's hypothesis H
1210:Manuscripta Mathematica
584:
557:is not prime (that is,
379:Fermat's little theorem
1162:
1116:
1071:
1025:
976:
902:
857:
783:
738:
702:
657:
540:
486:
433:
360:
311:
266:
139:Equivalent formulation
118:
1526:Waring's prime number
1163:
1090:
1072:
999:
977:
876:
858:
757:
739:
676:
658:
541:
487:
434:
361:
285:
267:
119:
33:Agoh–Giuga conjecture
18:Agoh-Giuga conjecture
1087:
996:
873:
866:and for p=2 we have
754:
673:
604:
496:
446:
388:
282:
162:
64:
1491:Legendre's constant
1442:Elliott–Halberstam
1427:Chinese hypothesis
1223:10.1007/bf02570490
1158:
1067:
972:
853:
734:
653:
536:
482:
429:
356:
262:
127:It is named after
114:
1534:
1533:
1462:Landau's problems
571:Carmichael number
37:Bernoulli numbers
16:(Redirected from
1559:
1380:Hardy–Littlewood
1364:
1357:
1350:
1341:
1336:
1315:
1313:
1312:
1306:
1300:. Archived from
1275:
1255:
1234:
1203:
1167:
1165:
1164:
1159:
1154:
1132:
1131:
1115:
1104:
1076:
1074:
1073:
1068:
1066:
1041:
1040:
1024:
1013:
981:
979:
978:
973:
968:
946:
945:
918:
917:
901:
890:
862:
860:
859:
854:
849:
827:
826:
799:
798:
782:
771:
743:
741:
740:
735:
730:
701:
690:
662:
660:
659:
654:
649:
591:Wilson's theorem
545:
543:
542:
537:
532:
491:
489:
488:
483:
438:
436:
435:
430:
428:
406:
405:
365:
363:
362:
357:
352:
327:
326:
310:
299:
271:
269:
268:
263:
261:
236:
235:
199:
198:
180:
179:
123:
121:
120:
115:
110:
85:
84:
47:postulates that
21:
1567:
1566:
1562:
1561:
1560:
1558:
1557:
1556:
1537:
1536:
1535:
1530:
1371:
1368:
1318:
1310:
1308:
1304:
1282:10.2307/2975213
1273:10.1.1.586.1424
1253:
1237:
1206:
1177:
1174:
1117:
1085:
1084:
1026:
994:
993:
931:
903:
871:
870:
812:
784:
752:
751:
671:
670:
602:
601:
587:
551:
494:
493:
444:
443:
391:
386:
385:
312:
280:
279:
221:
184:
165:
160:
159:
141:
70:
62:
61:
46:
23:
22:
15:
12:
11:
5:
1565:
1563:
1555:
1554:
1549:
1539:
1538:
1532:
1531:
1529:
1528:
1523:
1518:
1513:
1508:
1503:
1498:
1493:
1488:
1487:
1486:
1481:
1476:
1475:
1474:
1459:
1454:
1449:
1444:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1404:
1399:
1394:
1393:
1392:
1387:
1376:
1373:
1372:
1369:
1367:
1366:
1359:
1352:
1344:
1338:
1337:
1323:(in Italian).
1316:
1247:Borwein, P. B.
1243:Borwein, J. M.
1235:
1217:(4): 501–510.
1204:
1182:(in Italian).
1173:
1170:
1169:
1168:
1157:
1153:
1150:
1146:
1143:
1138:
1135:
1130:
1127:
1124:
1120:
1114:
1111:
1108:
1103:
1100:
1097:
1093:
1078:
1077:
1065:
1062:
1058:
1055:
1050:
1047:
1044:
1039:
1036:
1033:
1029:
1023:
1020:
1017:
1012:
1009:
1006:
1002:
983:
982:
971:
967:
964:
960:
957:
952:
949:
944:
941:
938:
934:
930:
927:
924:
921:
916:
913:
910:
906:
900:
897:
894:
889:
886:
883:
879:
864:
863:
852:
848:
845:
841:
838:
833:
830:
825:
822:
819:
815:
811:
808:
805:
802:
797:
794:
791:
787:
781:
778:
775:
770:
767:
764:
760:
745:
744:
733:
729:
726:
722:
719:
714:
711:
708:
705:
700:
697:
694:
689:
686:
683:
679:
664:
663:
652:
648:
645:
641:
638:
633:
630:
627:
624:
621:
618:
615:
612:
609:
586:
583:
550:
547:
535:
531:
528:
524:
521:
516:
513:
510:
507:
504:
501:
481:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
440:
439:
427:
424:
420:
417:
412:
409:
404:
401:
398:
394:
367:
366:
355:
351:
348:
344:
341:
336:
333:
330:
325:
322:
319:
315:
309:
306:
303:
298:
295:
292:
288:
273:
272:
260:
257:
253:
250:
245:
242:
239:
234:
231:
228:
224:
220:
217:
214:
211:
208:
205:
202:
197:
194:
191:
187:
183:
178:
175:
172:
168:
149:Giuseppe Giuga
140:
137:
133:Giuseppe Giuga
125:
124:
113:
109:
106:
102:
99:
94:
91:
88:
83:
80:
77:
73:
69:
56:if and only if
42:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1564:
1553:
1550:
1548:
1545:
1544:
1542:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1489:
1485:
1482:
1480:
1477:
1473:
1470:
1469:
1468:
1465:
1464:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1447:Firoozbakht's
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1408:
1405:
1403:
1400:
1398:
1395:
1391:
1388:
1386:
1383:
1382:
1381:
1378:
1377:
1374:
1365:
1360:
1358:
1353:
1351:
1346:
1345:
1342:
1334:
1330:
1326:
1322:
1317:
1307:on 2005-05-31
1303:
1299:
1295:
1291:
1287:
1283:
1279:
1274:
1269:
1265:
1261:
1260:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1212:
1211:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1176:
1175:
1171:
1155:
1148:
1144:
1136:
1133:
1128:
1125:
1122:
1118:
1112:
1109:
1106:
1101:
1098:
1095:
1091:
1083:
1082:
1081:
1060:
1056:
1048:
1045:
1042:
1037:
1034:
1031:
1027:
1021:
1018:
1015:
1010:
1007:
1004:
1000:
992:
991:
990:
988:
969:
962:
958:
950:
947:
942:
939:
936:
928:
925:
919:
914:
911:
908:
904:
898:
895:
892:
887:
884:
881:
877:
869:
868:
867:
850:
843:
839:
831:
828:
823:
820:
817:
809:
806:
800:
795:
792:
789:
785:
779:
776:
773:
768:
765:
762:
758:
750:
749:
748:
731:
724:
720:
712:
709:
706:
703:
698:
695:
692:
687:
684:
681:
677:
669:
668:
667:
650:
643:
639:
631:
628:
625:
622:
616:
613:
610:
600:
599:
598:
596:
592:
582:
580:
576:
572:
568:
564:
560:
556:
548:
546:
533:
526:
522:
514:
511:
508:
505:
502:
499:
479:
476:
473:
470:
467:
464:
461:
458:
455:
452:
449:
422:
418:
410:
407:
402:
399:
396:
392:
384:
383:
382:
380:
376:
372:
353:
346:
342:
334:
331:
328:
323:
320:
317:
313:
307:
304:
301:
296:
293:
290:
286:
278:
277:
276:
255:
251:
243:
240:
237:
232:
229:
226:
218:
215:
212:
206:
203:
200:
195:
192:
189:
185:
181:
176:
173:
170:
166:
158:
157:
156:
154:
150:
146:
138:
136:
134:
130:
111:
104:
100:
92:
89:
86:
81:
78:
75:
71:
67:
60:
59:
58:
57:
54:
50:
45:
41:
38:
34:
30:
29:number theory
19:
1412:Bateman–Horn
1396:
1324:
1320:
1309:. Retrieved
1302:the original
1266:(1): 40–50.
1263:
1257:
1214:
1208:
1183:
1179:
1079:
986:
984:
865:
746:
665:
594:
588:
578:
575:Giuga number
566:
558:
554:
552:
441:
381:states that
374:
370:
368:
274:
152:
145:Takashi Agoh
142:
129:Takashi Agoh
126:
53:prime number
48:
43:
39:
32:
26:
1506:Oppermann's
1452:Gilbreath's
1422:Bunyakovsky
1239:Borwein, D.
1186:: 511–518.
1541:Categories
1511:Polignac's
1484:Twin prime
1479:Legendre's
1467:Goldbach's
1397:Agoh–Giuga
1311:2005-05-29
1298:0860.11003
1231:0845.11004
1200:0045.01801
1172:References
377:is prime,
1496:Lemoine's
1437:Dickson's
1417:Brocard's
1402:Andrica's
1333:1720-9668
1268:CiteSeerX
1192:0375-9164
1134:≡
1126:−
1110:−
1092:∏
1046:−
1043:≡
1035:−
1019:−
1001:∑
948:≡
940:−
926:−
920:≡
912:−
896:−
878:∏
829:≡
821:−
807:−
801:≡
793:−
777:−
759:∏
710:−
707:≡
696:−
678:∏
629:−
626:≡
614:−
563:composite
512:−
509:≡
503:−
477:−
468:…
408:≡
400:−
332:−
329:≡
321:−
305:−
287:∑
241:−
238:≡
230:−
216:−
204:⋯
193:−
174:−
90:−
87:≡
79:−
1501:Mersenne
1432:Cramér's
1457:Grimm's
1407:Artin's
1290:2975213
35:on the
1331:
1296:
1288:
1270:
1229:
1198:
1190:
573:and a
549:Status
1516:Pólya
1305:(PDF)
1286:JSTOR
1254:(PDF)
51:is a
1472:weak
1329:ISSN
1188:ISSN
1080:and
442:for
131:and
31:the
1390:2nd
1385:1st
1294:Zbl
1278:doi
1264:103
1227:Zbl
1219:doi
1196:Zbl
1145:mod
1057:mod
959:mod
840:mod
721:mod
640:mod
561:is
523:mod
419:mod
343:mod
252:mod
101:mod
27:In
1543::
1327:.
1325:68
1292:.
1284:.
1276:.
1262:.
1256:.
1245:;
1241:;
1225:.
1215:87
1213:.
1194:.
1184:83
135:.
1363:e
1356:t
1349:v
1335:.
1314:.
1280::
1233:.
1221::
1202:.
1156:.
1152:)
1149:p
1142:(
1137:1
1129:1
1123:p
1119:i
1113:1
1107:p
1102:1
1099:=
1096:i
1064:)
1061:p
1054:(
1049:1
1038:1
1032:p
1028:i
1022:1
1016:p
1011:1
1008:=
1005:i
987:p
970:.
966:)
963:p
956:(
951:1
943:1
937:p
933:)
929:1
923:(
915:1
909:p
905:i
899:1
893:p
888:1
885:=
882:i
851:,
847:)
844:p
837:(
832:1
824:1
818:p
814:)
810:1
804:(
796:1
790:p
786:i
780:1
774:p
769:1
766:=
763:i
732:.
728:)
725:p
718:(
713:1
704:i
699:1
693:p
688:1
685:=
682:i
651:,
647:)
644:p
637:(
632:1
623:!
620:)
617:1
611:p
608:(
595:p
579:n
567:n
559:n
555:n
534:.
530:)
527:p
520:(
515:1
506:1
500:p
480:1
474:p
471:,
465:,
462:2
459:,
456:1
453:=
450:a
426:)
423:p
416:(
411:1
403:1
397:p
393:a
375:p
371:p
354:.
350:)
347:p
340:(
335:1
324:1
318:p
314:i
308:1
302:p
297:1
294:=
291:i
259:)
256:p
249:(
244:1
233:1
227:p
223:)
219:1
213:p
210:(
207:+
201:+
196:1
190:p
186:2
182:+
177:1
171:p
167:1
153:p
112:.
108:)
105:p
98:(
93:1
82:1
76:p
72:B
68:p
49:p
44:k
40:B
20:)
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