530:
371:
83:
announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600. As of April 14, 2014, one year after Zhang's announcement, according to the
651:
432:
1023:
738:
226:
978:
941:
872:
783:
199:
407:
1387:
1577:
1572:
1432:
569:
1415:
1380:
1088:(1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways … )
1410:
1113:
1551:
744:
follows. It relies on some unproven assumptions so the conclusion remains a conjecture. The chance of a random odd prime
1467:
1373:
1289:
1067:
93:
1256:
1235:
1405:
217:
525:{\displaystyle C_{2}=\prod _{p\geq 3}{\frac {p(p-2)}{(p-1)^{2}}}\approx 0.660161815846869573927812110014\dots }
1497:
1086:
Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières … "
80:
366:{\displaystyle \pi _{n}(x)\sim 2C_{n}{\frac {x}{(\ln x)^{2}}}\sim 2C_{n}\int _{2}^{x}{dt \over (\ln t)^{2}}}
1472:
1437:
983:
698:
1546:
1105:
946:
909:
1531:
1477:
1422:
848:
759:
1457:
1504:
1492:
1210:
1184:
1521:
1462:
1442:
1427:
168:
154:
684:. Twin primes have the same conjectured density as cousin primes, and half that of sexy primes.
1516:
1447:
1509:
1482:
1135:
108:
1526:
1487:
1332:
1158:
392:
33:
25:
1299:
1166:
8:
410:
1541:
1452:
1351:
741:
1348:
1329:
1285:
1109:
85:
1049: + 6, so the latter pair is conjectured twice as likely to both be prime.
59:
Although the conjecture has not yet been proven or disproven for any given value of
1295:
1162:
1144:
1317:
1277:
1154:
1099:
1149:
1130:
1566:
17:
1365:
116:
64:
48:
695:
increases the conjectured density compared to twin primes by a factor of
132:
1356:
1337:
1327:
157:
generalizes
Polignac's conjecture to cover all prime constellations.
68:
47:. In other words: There are infinitely many cases of two consecutive
40:
1025:
which transfers to the conjectured prime density. In the case of
96:
and its generalized form, the
Polymath project wiki states that
555:
multiplied by a number which depends on the odd prime factors
1346:
646:{\displaystyle C_{n}=C_{2}\prod _{q|n}{\frac {q-1}{q-2}}.}
988:
951:
914:
853:
764:
703:
1322:
Comptes Rendus des Séances de l'Académie des
Sciences
1033:
is a random number then 3 has chance 2/3 of dividing
986:
949:
912:
851:
762:
701:
572:
435:
395:
229:
171:
1211:"An old mathematical puzzle soon to be unraveled?"
1017:
972:
935:
866:
777:
732:
645:
524:
401:
365:
193:
131: = 6, it says there are infinitely many
535:where the product extends over all prime numbers
1564:
1185:"Unheralded Mathematician Bridges the Prime Gap"
1041: + 2, but only chance 1/3 of dividing
63:, in 2013 an important breakthrough was made by
1068:"Recherches nouvelles sur les nombres premiers"
1029: = 6, the argument simplifies to: If
756:+ 2 in a random "potential" twin prime pair is
92:has been reduced to 246. Further, assuming the
1276:
1381:
1318:Recherches nouvelles sur les nombres premiers
1395:
100:has been reduced to 12 and 6, respectively.
1065:
409:means that the quotient of two expressions
1388:
1374:
1097:
67:who proved that there are infinitely many
1182:
1148:
1101:Elementary number theory in nine chapters
1070:[New research on prime numbers].
1208:
220:says the asymptotic density is of form
115:= 4, it says there are infinitely many
1565:
1209:Augereau, Benjamin (15 January 2014).
1122:
160:
143: + 6) with no prime between
1369:
1347:
1328:
1128:
813:and consider a potential prime pair (
205:be the number of prime gaps of size
1018:{\displaystyle {\tfrac {q-1}{q-2}}}
733:{\displaystyle {\tfrac {q-1}{q-2}}}
13:
1578:Unsolved problems in number theory
79:< 70,000,000. Later that year,
14:
1589:
1284:, World Scientific, p. 313,
973:{\displaystyle {\tfrac {q-2}{q}}}
936:{\displaystyle {\tfrac {q-1}{q}}}
1183:Klarreich, Erica (19 May 2013).
687:Note that each odd prime factor
517:0.660161815846869573927812110014
1573:Conjectures about prime numbers
867:{\displaystyle {\tfrac {1}{q}}}
778:{\displaystyle {\tfrac {2}{q}}}
1270:
1249:
1228:
1202:
1176:
1091:
1059:
890:, divided by the chance that (
605:
501:
488:
483:
471:
351:
338:
284:
271:
246:
240:
188:
182:
1:
1310:
1280:; Diamond, Harold G. (2004),
1257:"Bounded gaps between primes"
1236:"Bounded gaps between primes"
1131:"Bounded gaps between primes"
886:) being free from the factor
94:Elliott–Halberstam conjecture
845:, and the chance of that is
39:, there are infinitely many
7:
1150:10.4007/annals.2014.179.3.7
805: − 1. Now assume
426:is the twin prime constant
218:Hardy–Littlewood conjecture
194:{\displaystyle \pi _{n}(x)}
10:
1594:
1333:"de Polignac's Conjecture"
1106:Cambridge University Press
1401:
1098:Tattersall, J.J. (2005),
1396:Prime number conjectures
1066:de Polignac, A. (1849).
1052:
1547:Schinzel's hypothesis H
1172:(subscription required)
1316:Alphonse de Polignac,
1282:Analytic Number Theory
1129:Zhang, Yitang (2014).
1019:
974:
937:
868:
779:
734:
647:
526:
403:
367:
195:
1552:Waring's prime number
1136:Annals of Mathematics
1020:
975:
938:
869:
780:
735:
648:
527:
417:approaches infinity.
404:
402:{\displaystyle \sim }
368:
196:
127: + 4). For
109:twin prime conjecture
86:Polymath project wiki
22:Polignac's conjecture
1352:"k-Tuple Conjecture"
984:
947:
910:
849:
760:
699:
570:
433:
393:
227:
169:
155:Dickson's conjecture
28:in 1849 and states:
26:Alphonse de Polignac
1517:Legendre's constant
1189:Simons Science News
789:divides one of the
326:
161:Conjectured density
1468:Elliott–Halberstam
1453:Chinese hypothesis
1349:Weisstein, Eric W.
1330:Weisstein, Eric W.
1015:
1013:
970:
968:
933:
931:
864:
862:
775:
773:
742:heuristic argument
730:
728:
643:
613:
522:
464:
399:
363:
312:
191:
75:for some value of
1560:
1559:
1488:Landau's problems
1115:978-0-521-85014-8
1012:
967:
930:
874:. The chance of (
861:
772:
727:
638:
596:
511:
449:
385:is a function of
361:
294:
32:For any positive
1585:
1406:Hardy–Littlewood
1390:
1383:
1376:
1367:
1366:
1362:
1361:
1343:
1342:
1304:
1302:
1278:Bateman, Paul T.
1274:
1268:
1267:
1265:
1264:
1253:
1247:
1246:
1244:
1243:
1232:
1226:
1225:
1223:
1221:
1206:
1200:
1199:
1197:
1195:
1180:
1174:
1173:
1170:
1152:
1143:(3): 1121–1174.
1126:
1120:
1118:
1095:
1089:
1079:
1063:
1024:
1022:
1021:
1016:
1014:
1011:
1000:
989:
979:
977:
976:
971:
969:
963:
952:
942:
940:
939:
934:
932:
926:
915:
873:
871:
870:
865:
863:
854:
784:
782:
781:
776:
774:
765:
748:dividing either
739:
737:
736:
731:
729:
726:
715:
704:
652:
650:
649:
644:
639:
637:
626:
615:
612:
608:
595:
594:
582:
581:
531:
529:
528:
523:
512:
510:
509:
508:
486:
466:
463:
445:
444:
408:
406:
405:
400:
372:
370:
369:
364:
362:
360:
359:
358:
336:
328:
325:
320:
311:
310:
295:
293:
292:
291:
266:
264:
263:
239:
238:
200:
198:
197:
192:
181:
180:
151: + 6.
51:with difference
1593:
1592:
1588:
1587:
1586:
1584:
1583:
1582:
1563:
1562:
1561:
1556:
1397:
1394:
1313:
1308:
1307:
1292:
1275:
1271:
1262:
1260:
1255:
1254:
1250:
1241:
1239:
1234:
1233:
1229:
1219:
1217:
1207:
1203:
1193:
1191:
1181:
1177:
1171:
1127:
1123:
1116:
1096:
1092:
1064:
1060:
1055:
1001:
990:
987:
985:
982:
981:
953:
950:
948:
945:
944:
916:
913:
911:
908:
907:
906:, then becomes
902:) is free from
852:
850:
847:
846:
837:if and only if
763:
761:
758:
757:
716:
705:
702:
700:
697:
696:
683:
676:
669:
662:
627:
616:
614:
604:
600:
590:
586:
577:
573:
571:
568:
567:
554:
546:
504:
500:
487:
467:
465:
453:
440:
436:
434:
431:
430:
425:
394:
391:
390:
384:
354:
350:
337:
329:
327:
321:
316:
306:
302:
287:
283:
270:
265:
259:
255:
234:
230:
228:
225:
224:
176:
172:
170:
167:
166:
163:
107:= 2, it is the
12:
11:
5:
1591:
1581:
1580:
1575:
1558:
1557:
1555:
1554:
1549:
1544:
1539:
1534:
1529:
1524:
1519:
1514:
1513:
1512:
1507:
1502:
1501:
1500:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1430:
1425:
1420:
1419:
1418:
1413:
1402:
1399:
1398:
1393:
1392:
1385:
1378:
1370:
1364:
1363:
1344:
1325:
1312:
1309:
1306:
1305:
1290:
1269:
1248:
1227:
1201:
1175:
1121:
1114:
1090:
1080:From p. 400:
1072:Comptes rendus
1057:
1056:
1054:
1051:
1010:
1007:
1004:
999:
996:
993:
980:. This equals
966:
962:
959:
956:
929:
925:
922:
919:
860:
857:
829: divides
771:
768:
725:
722:
719:
714:
711:
708:
681:
674:
667:
660:
654:
653:
642:
636:
633:
630:
625:
622:
619:
611:
607:
603:
599:
593:
589:
585:
580:
576:
552:
544:
533:
532:
521:
518:
515:
507:
503:
499:
496:
493:
490:
485:
482:
479:
476:
473:
470:
462:
459:
456:
452:
448:
443:
439:
423:
398:
380:
374:
373:
357:
353:
349:
346:
343:
340:
335:
332:
324:
319:
315:
309:
305:
301:
298:
290:
286:
282:
279:
276:
273:
269:
262:
258:
254:
251:
248:
245:
242:
237:
233:
190:
187:
184:
179:
175:
162:
159:
57:
56:
9:
6:
4:
3:
2:
1590:
1579:
1576:
1574:
1571:
1570:
1568:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1515:
1511:
1508:
1506:
1503:
1499:
1496:
1495:
1494:
1491:
1490:
1489:
1486:
1484:
1481:
1479:
1476:
1474:
1473:Firoozbakht's
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1417:
1414:
1412:
1409:
1408:
1407:
1404:
1403:
1400:
1391:
1386:
1384:
1379:
1377:
1372:
1371:
1368:
1359:
1358:
1353:
1350:
1345:
1340:
1339:
1334:
1331:
1326:
1323:
1319:
1315:
1314:
1301:
1297:
1293:
1291:981-256-080-7
1287:
1283:
1279:
1273:
1258:
1252:
1237:
1231:
1216:
1212:
1205:
1190:
1186:
1179:
1168:
1164:
1160:
1156:
1151:
1146:
1142:
1138:
1137:
1132:
1125:
1117:
1111:
1107:
1103:
1102:
1094:
1087:
1083:
1077:
1074:(in French).
1073:
1069:
1062:
1058:
1050:
1048:
1044:
1040:
1036:
1032:
1028:
1008:
1005:
1002:
997:
994:
991:
964:
960:
957:
954:
927:
923:
920:
917:
905:
901:
898: +
897:
893:
889:
885:
882: +
881:
877:
858:
855:
844:
840:
836:
833: +
832:
828:
824:
821: +
820:
816:
812:
808:
804:
801: +
800:
796:
793:numbers from
792:
788:
769:
766:
755:
751:
747:
743:
723:
720:
717:
712:
709:
706:
694:
690:
685:
680:
673:
666:
659:
656:For example,
640:
634:
631:
628:
623:
620:
617:
609:
601:
597:
591:
587:
583:
578:
574:
566:
565:
564:
562:
558:
551:
547:
540:
538:
519:
516:
513:
505:
497:
494:
491:
480:
477:
474:
468:
460:
457:
454:
450:
446:
441:
437:
429:
428:
427:
422:
418:
416:
412:
396:
388:
383:
379:
355:
347:
344:
341:
333:
330:
322:
317:
313:
307:
303:
299:
296:
288:
280:
277:
274:
267:
260:
256:
252:
249:
243:
235:
231:
223:
222:
221:
219:
214:
212:
208:
204:
185:
177:
173:
158:
156:
152:
150:
146:
142:
138:
134:
130:
126:
122:
118:
117:cousin primes
114:
110:
106:
101:
99:
95:
91:
87:
82:
81:James Maynard
78:
74:
70:
66:
62:
54:
50:
49:prime numbers
46:
42:
38:
35:
31:
30:
29:
27:
23:
19:
18:number theory
1536:
1438:Bateman–Horn
1355:
1336:
1321:
1281:
1272:
1261:. Retrieved
1251:
1240:. Retrieved
1230:
1218:. Retrieved
1214:
1204:
1192:. Retrieved
1188:
1178:
1140:
1134:
1124:
1100:
1093:
1085:
1081:
1075:
1071:
1061:
1046:
1042:
1038:
1034:
1030:
1026:
903:
899:
895:
891:
887:
883:
879:
875:
842:
838:
834:
830:
826:
822:
818:
814:
810:
806:
802:
798:
794:
790:
786:
753:
749:
745:
692:
688:
686:
678:
671:
664:
657:
655:
560:
556:
549:
542:
541:
536:
534:
420:
419:
414:
386:
381:
377:
375:
215:
210:
206:
202:
164:
153:
148:
144:
140:
136:
128:
124:
120:
112:
104:
102:
97:
89:
76:
72:
65:Yitang Zhang
60:
58:
52:
44:
36:
24:was made by
21:
15:
1532:Oppermann's
1478:Gilbreath's
1448:Bunyakovsky
1220:10 February
943:divided by
133:sexy primes
34:even number
1567:Categories
1537:Polignac's
1510:Twin prime
1505:Legendre's
1493:Goldbach's
1423:Agoh–Giuga
1311:References
1300:1074.11001
1263:2014-02-21
1259:. Polymath
1242:2014-03-27
1238:. Polymath
1167:1290.11128
1078:: 397–401.
216:The first
69:prime gaps
41:prime gaps
1522:Lemoine's
1463:Dickson's
1443:Brocard's
1428:Andrica's
1357:MathWorld
1338:MathWorld
1084:Théorème.
1006:−
995:−
958:−
921:−
721:−
710:−
632:−
621:−
598:∏
520:…
514:≈
495:−
478:−
458:≥
451:∏
397:∼
345:
314:∫
297:∼
278:
250:∼
232:π
201:for even
174:π
147:and
1527:Mersenne
1458:Cramér's
1215:Phys.org
1119:, p. 112
841:divides
809:divides
785:, since
411:tends to
71:of size
43:of size
1483:Grimm's
1433:Artin's
1159:3171761
878:,
817:,
139:,
123:,
1324:(1849)
1298:
1288:
1194:21 May
1165:
1157:
1112:
389:, and
376:where
209:below
111:. For
1542:Pólya
1053:Notes
539:≥ 3.
413:1 as
1498:weak
1286:ISBN
1222:2014
1196:2013
1110:ISBN
1045:and
825:).
740:. A
670:and
165:Let
103:For
1416:2nd
1411:1st
1296:Zbl
1163:Zbl
1145:doi
1141:179
1082:"1
1037:or
797:to
752:or
691:of
677:= 2
559:of
548:is
16:In
1569::
1354:.
1335:.
1320:.
1294:,
1213:.
1187:.
1161:.
1155:MR
1153:.
1139:.
1133:.
1108:,
1104:,
1076:29
894:,
663:=
563::
342:ln
275:ln
213:.
88:,
20:,
1389:e
1382:t
1375:v
1360:.
1341:.
1303:.
1266:.
1245:.
1224:.
1198:.
1169:.
1147::
1047:a
1043:a
1039:a
1035:a
1031:a
1027:n
1009:2
1003:q
998:1
992:q
965:q
961:2
955:q
928:q
924:1
918:q
904:q
900:2
896:a
892:a
888:q
884:n
880:a
876:a
859:q
856:1
843:a
839:q
835:n
831:a
827:q
823:n
819:a
815:a
811:n
807:q
803:q
799:a
795:a
791:q
787:q
770:q
767:2
754:a
750:a
746:q
724:2
718:q
713:1
707:q
693:n
689:q
682:2
679:C
675:6
672:C
668:2
665:C
661:4
658:C
641:.
635:2
629:q
624:1
618:q
610:n
606:|
602:q
592:2
588:C
584:=
579:n
575:C
561:n
557:q
553:2
550:C
545:n
543:C
537:p
506:2
502:)
498:1
492:p
489:(
484:)
481:2
475:p
472:(
469:p
461:3
455:p
447:=
442:2
438:C
424:2
421:C
415:x
387:n
382:n
378:C
356:2
352:)
348:t
339:(
334:t
331:d
323:x
318:2
308:n
304:C
300:2
289:2
285:)
281:x
272:(
268:x
261:n
257:C
253:2
247:)
244:x
241:(
236:n
211:x
207:n
203:n
189:)
186:x
183:(
178:n
149:p
145:p
141:p
137:p
135:(
129:n
125:p
121:p
119:(
113:n
105:n
98:n
90:n
77:n
73:n
61:n
55:.
53:n
45:n
37:n
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